<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@vertical+block@ps2" data-block-type="vertical" data-runtime-version="1" data-runtime-class="LmsRuntime" data-has-score="False" data-request-token="f70965bc0fc811efa5d10e90929de005" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Problem set 2</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@html+block@ps2_hashpdf">
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{"xmodule-type": "HTMLModule"}
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<p>You can download this pdf file from <a href="/assets/courseware/v1/badcad90a4023ce696929bcd6b7eb48c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_ps2.pdf">here</a></p><iframe src="/assets/courseware/v1/badcad90a4023ce696929bcd6b7eb48c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_ps2.pdf" width="100%" height="1000px"> </iframe>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@vertical+block@ps2-checkvertical" data-block-type="vertical" data-runtime-version="1" data-runtime-class="LmsRuntime" data-has-score="False" data-request-token="f70965bc0fc811efa5d10e90929de005" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Pset 2 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@ps2-1-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-1-check" data-block-type="problem" data-runtime-version="1" data-runtime-class="LmsRuntime" data-has-score="True" data-request-token="f70965bc0fc811efa5d10e90929de005" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="False" data-init="XBlockToXModuleShim">
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<div id="problem_ps2-1-check" class="problems-wrapper" role="group"
aria-labelledby="ps2-1-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-1-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-1-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps2-1-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-1-check-problem-progress" tabindex="-1">
Problem 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-1-check-problem-progress"></div>
<div class="problem">
<div>
<p>
To check your answers put them in the appropriate box and click the &#8216;Check' button. The checker can do arithmetic. So, 3^2, [mathjaxinline]5/3[/mathjaxinline] and [mathjaxinline]3*4[/mathjaxinline] are okay. </p>
<p>
Unless otherwise stated the checker expects numerical answers to 3 significant digits. </p>
<p>
For formulas you can use R notation. Remember to use a * for multiplication, i.e. [mathjaxinline]3*x[/mathjaxinline] <b class="bf">NOT </b> [mathjaxinline]3x[/mathjaxinline]. </p>
<p>
The checker knows many of the standard functions, e.g. use exp(x) and ln(x) for the exponential and natural log functions. </p>
<p>
The formula checker expects a specific letter for the variables, so be sure to use the ones specified. </p>
<p><b class="bf">1(a)</b> Give the probabality of two females:<br/><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-1-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-1-check_2_1" id="input_ps2-1-check_2_1" aria-describedby="status_ps2-1-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-1-check_2_1"/>
<span class="status unanswered" id="status_ps2-1-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-1-check_2_1" class="answer"/>
<div id="display_ps2-1-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-1-check_2_1_dynamath" name="input_ps2-1-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bf">1(b)</b> Give the probabality of two males:<br/><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps2-1-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-1-check_3_1" id="input_ps2-1-check_3_1" aria-describedby="status_ps2-1-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-1-check_3_1"/>
<span class="status unanswered" id="status_ps2-1-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-1-check_3_1" class="answer"/>
<div id="display_ps2-1-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-1-check_3_1_dynamath" name="input_ps2-1-check_3_1_dynamath"/>
</div>
</div></div> </p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 1" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps2-1-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps2-1-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-1-check-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-1-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-1-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-2-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-2-check" data-block-type="problem" data-runtime-version="1" data-runtime-class="LmsRuntime" data-has-score="True" data-request-token="f70965bc0fc811efa5d10e90929de005" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps2-2-check" class="problems-wrapper" role="group"
aria-labelledby="ps2-2-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-2-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-2-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
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<h3 class="hd hd-3 problem-header" id="ps2-2-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-2-check-problem-progress" tabindex="-1">
Problem 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-2-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">2.</b> Give P(guilty | the witness saw a blue taxi):<br/><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-2-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-2-check_2_1" id="input_ps2-2-check_2_1" aria-describedby="status_ps2-2-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-2-check_2_1"/>
<span class="status unanswered" id="status_ps2-2-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-2-check_2_1" class="answer"/>
<div id="display_ps2-2-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-2-check_2_1_dynamath" name="input_ps2-2-check_2_1_dynamath"/>
</div>
</div></div> </p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 2" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps2-2-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps2-2-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-2-check-problem-title">
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<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-2-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-2-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-3-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-3-check" data-block-type="problem" data-runtime-version="1" data-runtime-class="LmsRuntime" data-has-score="True" data-request-token="f70965bc0fc811efa5d10e90929de005" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
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<div id="problem_ps2-3-check" class="problems-wrapper" role="group"
aria-labelledby="ps2-3-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-3-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-3-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
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data-content="
<h3 class="hd hd-3 problem-header" id="ps2-3-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-3-check-problem-progress" tabindex="-1">
Problem 3
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-3-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">3.</b> Give the expected sum of cards:<br/><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-3-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-3-check_2_1" id="input_ps2-3-check_2_1" aria-describedby="status_ps2-3-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-3-check_2_1"/>
<span class="status unanswered" id="status_ps2-3-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-3-check_2_1" class="answer"/>
<div id="display_ps2-3-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-3-check_2_1_dynamath" name="input_ps2-3-check_2_1_dynamath"/>
</div>
</div></div> </p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 3" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps2-3-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps2-3-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-3-check-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-3-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-3-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-4-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-4-check" data-block-type="problem" data-runtime-version="1" data-runtime-class="LmsRuntime" data-has-score="True" data-request-token="f70965bc0fc811efa5d10e90929de005" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
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<div id="problem_ps2-4-check" class="problems-wrapper" role="group"
aria-labelledby="ps2-4-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-4-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-4-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps2-4-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-4-check-problem-progress" tabindex="-1">
Problem 4
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-4-check-problem-progress"></div>
<div class="problem">
<div>
<p>
<b class="bf">4(a)</b>
<em>No answer checker given.</em>
</p>
<p><b class="bf">4(b)</b> Give [mathjaxinline]P(S=s | R=3)[/mathjaxinline]: </p>
<p>
For [mathjaxinline]s = 4[/mathjaxinline]: <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-4-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-4-check_2_1" id="input_ps2-4-check_2_1" aria-describedby="status_ps2-4-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-4-check_2_1"/>
<span class="status unanswered" id="status_ps2-4-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-4-check_2_1" class="answer"/>
<div id="display_ps2-4-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-4-check_2_1_dynamath" name="input_ps2-4-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p>
For [mathjaxinline]s = 6[/mathjaxinline]: <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps2-4-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-4-check_3_1" id="input_ps2-4-check_3_1" aria-describedby="status_ps2-4-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-4-check_3_1"/>
<span class="status unanswered" id="status_ps2-4-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-4-check_3_1" class="answer"/>
<div id="display_ps2-4-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-4-check_3_1_dynamath" name="input_ps2-4-check_3_1_dynamath"/>
</div>
</div></div> </p>
<p>
For [mathjaxinline]s = 8[/mathjaxinline]: <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps2-4-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-4-check_4_1" id="input_ps2-4-check_4_1" aria-describedby="status_ps2-4-check_4_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-4-check_4_1"/>
<span class="status unanswered" id="status_ps2-4-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-4-check_4_1" class="answer"/>
<div id="display_ps2-4-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-4-check_4_1_dynamath" name="input_ps2-4-check_4_1_dynamath"/>
</div>
</div></div> </p>
<p>
<b class="bf">4(c)</b>
<em>No answer checker given.</em>
</p>
<p>
<b class="bf">4(d)</b>
<em>No answer checker given.</em>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 4" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps2-4-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps2-4-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-4-check-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-4-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-4-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-5-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-5-check" data-block-type="problem" data-runtime-version="1" data-runtime-class="LmsRuntime" data-has-score="True" data-request-token="f70965bc0fc811efa5d10e90929de005" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps2-5-check" class="problems-wrapper" role="group"
aria-labelledby="ps2-5-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-5-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-5-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps2-5-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-5-check-problem-progress" tabindex="-1">
Problem 5
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-5-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">5.</b> Give [mathjaxinline]E[X][/mathjaxinline] as a function of [mathjaxinline]n[/mathjaxinline]. <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-5-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-5-check_2_1" id="input_ps2-5-check_2_1" aria-describedby="status_ps2-5-check_2_1" value="" class="math" size="70"/>
<span class="trailing_text" id="trailing_text_ps2-5-check_2_1"/>
<span class="status unanswered" id="status_ps2-5-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-5-check_2_1" class="answer"/>
<div id="display_ps2-5-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-5-check_2_1_dynamath" name="input_ps2-5-check_2_1_dynamath"/>
</div>
</div></div> </p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 5" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps2-5-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps2-5-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-5-check-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-5-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-5-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-5" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-6-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-6-check" data-block-type="problem" data-runtime-version="1" data-runtime-class="LmsRuntime" data-has-score="True" data-request-token="f70965bc0fc811efa5d10e90929de005" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="False" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps2-6-check" class="problems-wrapper" role="group"
aria-labelledby="ps2-6-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-6-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-6-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps2-6-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-6-check-problem-progress" tabindex="-1">
Problem 6
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps2-6-check-problem-progress"></div>
<div class="problem">
<div>
<p>
<b class="bf">6(a)</b>
<em>No answer checker given.</em>
</p>
<p>
<b class="bf">6(b)</b>
<em>No answer checker given.</em>
</p>
<p><b class="bf">6(c)</b> Give the average length of the longest run in your simulation.<br/><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-6-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-6-check_2_1" id="input_ps2-6-check_2_1" aria-describedby="status_ps2-6-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-6-check_2_1"/>
<span class="status unanswered" id="status_ps2-6-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-6-check_2_1" class="answer"/>
<div id="display_ps2-6-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-6-check_2_1_dynamath" name="input_ps2-6-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p>
<em>Because this is a simulation we have a fairly wide tolerance for different answers. But, if the checker doesn't like your answer run the simulation again and try the new answer. Be sure you are using at least 10000 trials.</em>
</p>
<p><b class="bf">6(d)</b> Give your estimated probability for the a run of 8 or more in 50 flips.<br/><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps2-6-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-6-check_3_1" id="input_ps2-6-check_3_1" aria-describedby="status_ps2-6-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-6-check_3_1"/>
<span class="status unanswered" id="status_ps2-6-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-6-check_3_1" class="answer"/>
<div id="display_ps2-6-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-6-check_3_1_dynamath" name="input_ps2-6-check_3_1_dynamath"/>
</div>
</div></div> </p>
<p>
<em>The comment in part (c) applies here also.</em>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 6" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps2-6-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps2-6-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-6-check-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-6-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps2-6-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>