<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@vertical+block@ps7" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">Problem set 7</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@html+block@ps7_hashpdf"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-block-type="html" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@html+block@ps7_hashpdf" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p>You can download this pdf file from <a href="/assets/courseware/v1/1b92c54e728fda172f75aacc5475ac7c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_ps7.pdf">here</a></p><iframe src="/assets/courseware/v1/1b92c54e728fda172f75aacc5475ac7c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_ps7.pdf" width="100%" height="1000px"> </iframe> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@vertical+block@ps7-checkvertical" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">Pset 7 problem checker</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-1-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-1-check" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps7-1-check" class="problems-wrapper" role="group" aria-labelledby="ps7-1-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-1-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-1-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps7-1-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-1-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 1 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-1-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; To check your answers put them in the appropriate box and click the &amp;#8216;Check&#39; button. The checker can do arithmetic. So, 3^2, [mathjaxinline]5/3[/mathjaxinline] and [mathjaxinline]3*4[/mathjaxinline] are okay. &lt;/p&gt; &lt;p&gt; Unless otherwise stated the checker expects numerical answers to 3 significant digits. &lt;/p&gt; &lt;p&gt; For formulas you can use R notation. Remember to use a * for multiplication, i.e. [mathjaxinline]3*x[/mathjaxinline] &lt;b class=&#34;bf&#34;&gt;NOT &lt;/b&gt; [mathjaxinline]3x[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; The checker knows many of the standard functions, e.g. use exp(x) and ln(x) for the exponential and natural log functions. &lt;/p&gt; &lt;p&gt; The formula checker expects a specific letter for the variables, so be sure to use the ones specified. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;1.&lt;/b&gt; Give your answers to at least 3 significant digits. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;(a)&lt;/b&gt; With sober Monty, what is the posterior probability that Shelby will win if she switches? &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-1-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-1-check_2_1&#34; id=&#34;input_ps7-1-check_2_1&#34; aria-describedby=&#34;status_ps7-1-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-1-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-1-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-1-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-1-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-1-check_2_1_dynamath&#34; name=&#34;input_ps7-1-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;(b)&lt;/b&gt; With drunk Monty, what is the posterior probability that Shelby will win if she switches? &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-1-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-1-check_3_1&#34; id=&#34;input_ps7-1-check_3_1&#34; aria-describedby=&#34;status_ps7-1-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-1-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-1-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-1-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-1-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-1-check_3_1_dynamath&#34; name=&#34;input_ps7-1-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;(c)&lt;/b&gt; With maybe drunk Monty, what is the posterior probability that Shelby will win if she switches? &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-1-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-1-check_4_1&#34; id=&#34;input_ps7-1-check_4_1&#34; aria-describedby=&#34;status_ps7-1-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-1-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-1-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-1-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-1-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-1-check_4_1_dynamath&#34; name=&#34;input_ps7-1-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 1&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps7-1-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps7-1-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-1-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-1-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-1-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2a-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2a-check" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps7-2a-check" class="problems-wrapper" role="group" aria-labelledby="ps7-2a-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2a-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2a-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps7-2a-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2a-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 2a &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2a-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;2(a).&lt;/b&gt; Give your answers to at least three significant digits. Remember MITx understands scientific notation, i.e. 0.000321 = 3.21e-4. &lt;/p&gt; &lt;p&gt; Give the prior predictive probability of a 7. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2a-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2a-check_2_1&#34; id=&#34;input_ps7-2a-check_2_1&#34; aria-describedby=&#34;status_ps7-2a-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2a-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2a-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2a-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2a-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2a-check_2_1_dynamath&#34; name=&#34;input_ps7-2a-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Let [mathjaxinline]D_4[/mathjaxinline] be the hypothesis the chosen die is four-sided and likewise for [mathjaxinline]D_6[/mathjaxinline] etc. Give the posterior probability of each hypothesis. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]D_4[/mathjaxinline]: &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2a-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2a-check_3_1&#34; id=&#34;input_ps7-2a-check_3_1&#34; aria-describedby=&#34;status_ps7-2a-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2a-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2a-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2a-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2a-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2a-check_3_1_dynamath&#34; name=&#34;input_ps7-2a-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]D_6[/mathjaxinline]: &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2a-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2a-check_4_1&#34; id=&#34;input_ps7-2a-check_4_1&#34; aria-describedby=&#34;status_ps7-2a-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2a-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2a-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2a-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2a-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2a-check_4_1_dynamath&#34; name=&#34;input_ps7-2a-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]D_8[/mathjaxinline]: &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2a-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2a-check_5_1&#34; id=&#34;input_ps7-2a-check_5_1&#34; aria-describedby=&#34;status_ps7-2a-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2a-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2a-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2a-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2a-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2a-check_5_1_dynamath&#34; name=&#34;input_ps7-2a-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]D_{12}[/mathjaxinline]: &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2a-check_6_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2a-check_6_1&#34; id=&#34;input_ps7-2a-check_6_1&#34; aria-describedby=&#34;status_ps7-2a-check_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2a-check_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2a-check_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2a-check_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2a-check_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2a-check_6_1_dynamath&#34; name=&#34;input_ps7-2a-check_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]D_{20}[/mathjaxinline]: &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 6&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2a-check_7_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2a-check_7_1&#34; id=&#34;input_ps7-2a-check_7_1&#34; aria-describedby=&#34;status_ps7-2a-check_7_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2a-check_7_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2a-check_7_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2a-check_7_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2a-check_7_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2a-check_7_1_dynamath&#34; name=&#34;input_ps7-2a-check_7_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 2a&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps7-2a-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps7-2a-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2a-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2a-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2a-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-2" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2b-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2b-check" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps7-2b-check" class="problems-wrapper" role="group" aria-labelledby="ps7-2b-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2b-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2b-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps7-2b-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2b-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 2b &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2b-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;2(b) (general [mathjaxinline]n[/mathjaxinline])&lt;/b&gt; To make your answers easier to enter, first give the total probability that the first [mathjaxinline]n[/mathjaxinline] rolls are all 7. We&#39;ll call this probability [mathjaxinline]T[/mathjaxinline]. Your answer should use the variable &lt;tt class=&#34;tt&#34;&gt;n&lt;/tt&gt;. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]T =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2b-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2b-check_2_1&#34; id=&#34;input_ps7-2b-check_2_1&#34; aria-describedby=&#34;status_ps7-2b-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2b-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2b-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2b-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2b-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2b-check_2_1_dynamath&#34; name=&#34;input_ps7-2b-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Now enter the poster probabilities for each type of die. You should use [mathjaxinline]T[/mathjaxinline] for the total probability of the data ([mathjaxinline]n[/mathjaxinline] sevens in a row). &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]P(D_4 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2b-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2b-check_3_1&#34; id=&#34;input_ps7-2b-check_3_1&#34; aria-describedby=&#34;status_ps7-2b-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2b-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2b-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2b-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2b-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2b-check_3_1_dynamath&#34; name=&#34;input_ps7-2b-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]P(D_6 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2b-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2b-check_4_1&#34; id=&#34;input_ps7-2b-check_4_1&#34; aria-describedby=&#34;status_ps7-2b-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2b-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2b-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2b-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2b-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2b-check_4_1_dynamath&#34; name=&#34;input_ps7-2b-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]P(D_8 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2b-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2b-check_5_1&#34; id=&#34;input_ps7-2b-check_5_1&#34; aria-describedby=&#34;status_ps7-2b-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2b-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2b-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2b-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2b-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2b-check_5_1_dynamath&#34; name=&#34;input_ps7-2b-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]P(D_{12} |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2b-check_6_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2b-check_6_1&#34; id=&#34;input_ps7-2b-check_6_1&#34; aria-describedby=&#34;status_ps7-2b-check_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2b-check_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2b-check_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2b-check_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2b-check_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2b-check_6_1_dynamath&#34; name=&#34;input_ps7-2b-check_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]P(D_{20} |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 6&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2b-check_7_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2b-check_7_1&#34; id=&#34;input_ps7-2b-check_7_1&#34; aria-describedby=&#34;status_ps7-2b-check_7_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2b-check_7_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2b-check_7_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2b-check_7_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2b-check_7_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2b-check_7_1_dynamath&#34; name=&#34;input_ps7-2b-check_7_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 2b&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps7-2b-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps7-2b-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2b-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2b-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2b-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-3" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2blimit-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2blimit-check" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps7-2blimit-check" class="problems-wrapper" role="group" aria-labelledby="ps7-2blimit-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2blimit-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2blimit-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps7-2blimit-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2blimit-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 2blimit &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2blimit-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;2(b) (in the limit)&lt;/b&gt; Give the posterior probability of each die in the limit as [mathjaxinline]n[/mathjaxinline] goes to infinity. We expect numerical answers that don&#39;t depend on [mathjaxinline]n[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle \lim _{n\to \infty }P(D_4 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2blimit-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2blimit-check_2_1&#34; id=&#34;input_ps7-2blimit-check_2_1&#34; aria-describedby=&#34;status_ps7-2blimit-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2blimit-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2blimit-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2blimit-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2blimit-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2blimit-check_2_1_dynamath&#34; name=&#34;input_ps7-2blimit-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle \lim _{n\to \infty }P(D_6 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2blimit-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2blimit-check_3_1&#34; id=&#34;input_ps7-2blimit-check_3_1&#34; aria-describedby=&#34;status_ps7-2blimit-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2blimit-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2blimit-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2blimit-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2blimit-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2blimit-check_3_1_dynamath&#34; name=&#34;input_ps7-2blimit-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle \lim _{n\to \infty }P(D_8 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2blimit-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2blimit-check_4_1&#34; id=&#34;input_ps7-2blimit-check_4_1&#34; aria-describedby=&#34;status_ps7-2blimit-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2blimit-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2blimit-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2blimit-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2blimit-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2blimit-check_4_1_dynamath&#34; name=&#34;input_ps7-2blimit-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle \lim _{n\to \infty }P(D_{12} |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2blimit-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2blimit-check_5_1&#34; id=&#34;input_ps7-2blimit-check_5_1&#34; aria-describedby=&#34;status_ps7-2blimit-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2blimit-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2blimit-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2blimit-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2blimit-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2blimit-check_5_1_dynamath&#34; name=&#34;input_ps7-2blimit-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle \lim _{n\to \infty }P(D_{20} |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2blimit-check_6_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2blimit-check_6_1&#34; id=&#34;input_ps7-2blimit-check_6_1&#34; aria-describedby=&#34;status_ps7-2blimit-check_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2blimit-check_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2blimit-check_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2blimit-check_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2blimit-check_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2blimit-check_6_1_dynamath&#34; name=&#34;input_ps7-2blimit-check_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 2blimit&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps7-2blimit-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps7-2blimit-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2blimit-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2blimit-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2blimit-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-4" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2c-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2c-check" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps7-2c-check" class="problems-wrapper" role="group" aria-labelledby="ps7-2c-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2c-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2c-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps7-2c-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2c-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 2c &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2c-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;2(c)&lt;/b&gt; We&#39;ll tell you that we can break the outcomes into three sets. Within each set the outcomes have equal probability. &lt;/p&gt; &lt;p&gt; You should give each set as a comma separated list of integers. &lt;/p&gt; &lt;p&gt; Give the set where each outcome has the greatest probability. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2c-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2c-check_2_1&#34; id=&#34;input_ps7-2c-check_2_1&#34; aria-describedby=&#34;status_ps7-2c-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2c-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2c-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2c-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2c-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2c-check_2_1_dynamath&#34; name=&#34;input_ps7-2c-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Give the set where each outcome has the second greatest probability. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2c-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2c-check_3_1&#34; id=&#34;input_ps7-2c-check_3_1&#34; aria-describedby=&#34;status_ps7-2c-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2c-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2c-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2c-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2c-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2c-check_3_1_dynamath&#34; name=&#34;input_ps7-2c-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Give the set where each outcome has the smallest probability. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2c-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2c-check_4_1&#34; id=&#34;input_ps7-2c-check_4_1&#34; aria-describedby=&#34;status_ps7-2c-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2c-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2c-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2c-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2c-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2c-check_4_1_dynamath&#34; name=&#34;input_ps7-2c-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 2c&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps7-2c-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps7-2c-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2c-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2c-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2c-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-5" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2d-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2d-check" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps7-2d-check" class="problems-wrapper" role="group" aria-labelledby="ps7-2d-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2d-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2d-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps7-2d-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2d-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 2d &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2d-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;2(d)&lt;/b&gt; For this problem, you can use the letter T to stand for the same total probability found in part (b). This may simplify entering your formulas. &lt;/p&gt; &lt;p&gt; There are 20 probabilities for this problem. We will only check five of them here. &lt;/p&gt; &lt;p&gt; Give the formula in terms of n and T for each of the probabilities indicated. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]P(x_{n+1} = 4 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2d-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2d-check_2_1&#34; id=&#34;input_ps7-2d-check_2_1&#34; aria-describedby=&#34;status_ps7-2d-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;65&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2d-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2d-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2d-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2d-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2d-check_2_1_dynamath&#34; name=&#34;input_ps7-2d-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]P(x_{n+1} = 6 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2d-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2d-check_3_1&#34; id=&#34;input_ps7-2d-check_3_1&#34; aria-describedby=&#34;status_ps7-2d-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;65&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2d-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2d-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2d-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2d-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2d-check_3_1_dynamath&#34; name=&#34;input_ps7-2d-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]P(x_{n+1} = 8 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2d-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2d-check_4_1&#34; id=&#34;input_ps7-2d-check_4_1&#34; aria-describedby=&#34;status_ps7-2d-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;65&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2d-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2d-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2d-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2d-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2d-check_4_1_dynamath&#34; name=&#34;input_ps7-2d-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]P(x_{n+1} = 12 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2d-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2d-check_5_1&#34; id=&#34;input_ps7-2d-check_5_1&#34; aria-describedby=&#34;status_ps7-2d-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;65&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2d-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2d-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2d-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2d-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2d-check_5_1_dynamath&#34; name=&#34;input_ps7-2d-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]P(x_{n+1} = 20 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2d-check_6_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2d-check_6_1&#34; id=&#34;input_ps7-2d-check_6_1&#34; aria-describedby=&#34;status_ps7-2d-check_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;65&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2d-check_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2d-check_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2d-check_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2d-check_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2d-check_6_1_dynamath&#34; name=&#34;input_ps7-2d-check_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 2d&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps7-2d-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps7-2d-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2d-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2d-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2d-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-6" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2e-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2e-check" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps7-2e-check" class="problems-wrapper" role="group" aria-labelledby="ps7-2e-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2e-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2e-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps7-2e-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2e-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 2e &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-2e-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;2(e)&lt;/b&gt; As in part (d), there are 20 probabilities to check. We will only check 5 of them here. &lt;/p&gt; &lt;p&gt; Give each of the probabilities indicated to 3 significant digits. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle \lim _{n\to \infty } P(x_{n+1} = 4 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2e-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2e-check_2_1&#34; id=&#34;input_ps7-2e-check_2_1&#34; aria-describedby=&#34;status_ps7-2e-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2e-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2e-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2e-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2e-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2e-check_2_1_dynamath&#34; name=&#34;input_ps7-2e-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle \lim _{n\to \infty } P(x_{n+1} = 6 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2e-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2e-check_3_1&#34; id=&#34;input_ps7-2e-check_3_1&#34; aria-describedby=&#34;status_ps7-2e-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2e-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2e-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2e-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2e-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2e-check_3_1_dynamath&#34; name=&#34;input_ps7-2e-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle \lim _{n\to \infty } P(x_{n+1} = 8 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2e-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2e-check_4_1&#34; id=&#34;input_ps7-2e-check_4_1&#34; aria-describedby=&#34;status_ps7-2e-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2e-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2e-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2e-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2e-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2e-check_4_1_dynamath&#34; name=&#34;input_ps7-2e-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle \lim _{n\to \infty } P(x_{n+1} = 12 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2e-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2e-check_5_1&#34; id=&#34;input_ps7-2e-check_5_1&#34; aria-describedby=&#34;status_ps7-2e-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2e-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2e-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2e-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2e-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2e-check_5_1_dynamath&#34; name=&#34;input_ps7-2e-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle \lim _{n\to \infty } P(x_{n+1} = 20 |\text {data}) =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-2e-check_6_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-2e-check_6_1&#34; id=&#34;input_ps7-2e-check_6_1&#34; aria-describedby=&#34;status_ps7-2e-check_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-2e-check_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-2e-check_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-2e-check_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-2e-check_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-2e-check_6_1_dynamath&#34; name=&#34;input_ps7-2e-check_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 2e&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps7-2e-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps7-2e-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2e-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2e-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-2e-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-7" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-3-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-3-check" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps7-3-check" class="problems-wrapper" role="group" aria-labelledby="ps7-3-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-3-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-3-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps7-3-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-3-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 3 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-3-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;3.&lt;/b&gt; Give all answers to at least 3 significant digits. If your answer is an exact fraction, e.g. 11/19, you can enter it that way. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;3(a)&lt;/b&gt; Give the prior odds of choosing a 0.3 coin. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-3-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-3-check_2_1&#34; id=&#34;input_ps7-3-check_2_1&#34; aria-describedby=&#34;status_ps7-3-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-3-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-3-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-3-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-3-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-3-check_2_1_dynamath&#34; name=&#34;input_ps7-3-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Give the prior odds of choosing a 0.7 coin. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-3-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-3-check_3_1&#34; id=&#34;input_ps7-3-check_3_1&#34; aria-describedby=&#34;status_ps7-3-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-3-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-3-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-3-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-3-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-3-check_3_1_dynamath&#34; name=&#34;input_ps7-3-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;3(b)&lt;/b&gt; Give the prior odds of flipping a heads. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-3-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-3-check_4_1&#34; id=&#34;input_ps7-3-check_4_1&#34; aria-describedby=&#34;status_ps7-3-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-3-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-3-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-3-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-3-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-3-check_4_1_dynamath&#34; name=&#34;input_ps7-3-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;3(c)&lt;/b&gt; Give the posterior odds you chose a 0.3 coin. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-3-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-3-check_5_1&#34; id=&#34;input_ps7-3-check_5_1&#34; aria-describedby=&#34;status_ps7-3-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-3-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-3-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-3-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-3-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-3-check_5_1_dynamath&#34; name=&#34;input_ps7-3-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Give the posterior odds you chose a 0.7 coin. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-3-check_6_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-3-check_6_1&#34; id=&#34;input_ps7-3-check_6_1&#34; aria-describedby=&#34;status_ps7-3-check_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-3-check_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-3-check_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-3-check_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-3-check_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-3-check_6_1_dynamath&#34; name=&#34;input_ps7-3-check_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;3(d)&lt;/b&gt; Give the posterior predictive odds of heads on the second flip. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 6&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-3-check_7_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-3-check_7_1&#34; id=&#34;input_ps7-3-check_7_1&#34; aria-describedby=&#34;status_ps7-3-check_7_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-3-check_7_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-3-check_7_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-3-check_7_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-3-check_7_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-3-check_7_1_dynamath&#34; name=&#34;input_ps7-3-check_7_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 3&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps7-3-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps7-3-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-3-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-3-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-3-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-8" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-4-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-4-check" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps7-4-check" class="problems-wrapper" role="group" aria-labelledby="ps7-4-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-4-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-4-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps7-4-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-4-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 4 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-4-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; &lt;b class=&#34;bf&#34;&gt;4(a)&lt;/b&gt; &lt;em&gt;No answer checker given.&lt;/em&gt; &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bf&#34;&gt;4(b)&lt;/b&gt; &lt;em&gt;No answer checker given.&lt;/em&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 4&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps7-4-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps7-4-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-4-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-4-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-4-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-9" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-5-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-5-check" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps7-5-check" class="problems-wrapper" role="group" aria-labelledby="ps7-5-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-5-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-5-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps7-5-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-5-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 5 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-5-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;5.&lt;/b&gt; Give the posterior odds Oscar is filling in for Emmy. (Give to at least 3 significant digits.) &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-5-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-5-check_2_1&#34; id=&#34;input_ps7-5-check_2_1&#34; aria-describedby=&#34;status_ps7-5-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-5-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-5-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-5-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-5-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-5-check_2_1_dynamath&#34; name=&#34;input_ps7-5-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 5&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps7-5-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps7-5-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-5-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-5-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-5-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-10" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-6-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-6-check" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps7-6-check" class="problems-wrapper" role="group" aria-labelledby="ps7-6-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-6-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-6-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps7-6-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-6-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 6 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-6-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Give your answers correct to 3 significant digits. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;6(a)&lt;/b&gt; Give the mean of the posterior pdf. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-6-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-6-check_2_1&#34; id=&#34;input_ps7-6-check_2_1&#34; aria-describedby=&#34;status_ps7-6-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-6-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-6-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-6-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-6-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-6-check_2_1_dynamath&#34; name=&#34;input_ps7-6-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Give the variance of the posterior pdf. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-6-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-6-check_3_1&#34; id=&#34;input_ps7-6-check_3_1&#34; aria-describedby=&#34;status_ps7-6-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-6-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-6-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-6-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-6-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-6-check_3_1_dynamath&#34; name=&#34;input_ps7-6-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bf&#34;&gt;6(b)&lt;/b&gt; &lt;em&gt;No answer checker given.&lt;/em&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;6(c)&lt;/b&gt; Give the posterior expected values of their IQs. (Give to at least 3 significant digits.) &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;R. Vard: &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-6-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-6-check_4_1&#34; id=&#34;input_ps7-6-check_4_1&#34; aria-describedby=&#34;status_ps7-6-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-6-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-6-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-6-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-6-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-6-check_4_1_dynamath&#34; name=&#34;input_ps7-6-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;M. I. Taft: &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-6-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-6-check_5_1&#34; id=&#34;input_ps7-6-check_5_1&#34; aria-describedby=&#34;status_ps7-6-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-6-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-6-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-6-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-6-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-6-check_5_1_dynamath&#34; name=&#34;input_ps7-6-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 6&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps7-6-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps7-6-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-6-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-6-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-6-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-11" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-7-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-7-check" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-request-token="ac42141a075311f095dc0e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps7-7-check" class="problems-wrapper" role="group" aria-labelledby="ps7-7-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-7-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-7-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps7-7-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-7-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 7 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps7-7-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;7(a)&lt;/b&gt; Give, as a comma separated list, the posterior odds the chosen die is 4-sided after rolls 1, 2, 3, 4, 5. You can enter these as exact fractions or as decimals to at least 3 significant digits. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps7-7-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps7-7-check_2_1&#34; id=&#34;input_ps7-7-check_2_1&#34; aria-describedby=&#34;status_ps7-7-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps7-7-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps7-7-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps7-7-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps7-7-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps7-7-check_2_1_dynamath&#34; name=&#34;input_ps7-7-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bf&#34;&gt;7(b)&lt;/b&gt; &lt;em&gt;No answer checker given.&lt;/em&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 7&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps7-7-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps7-7-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-7-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-7-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps7-7-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> </div> </div>