<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@26a9c5c12fea40e5afee3c6a97058058">
<h2 class="hd hd-2 unit-title">Introduction</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@c88dc6f15f0248beb11e9604c52cb712">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="html" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@c88dc6f15f0248beb11e9604c52cb712">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Read <a href="/assets/courseware/v1/2acec82901487d019fdf70875a1d11de/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session30.pdf" target="[object Object]">Chapter 17.7–17.8 (PDF)</a> of <em>Mathematics for Computer Science</em> for 4.3 Independence & Causality</p>
<p>View the <a href="/assets/courseware/v1/0faabf617749c261be6d060d87ff1778/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp30.pdf" target="[object Object]">Section 4.3 In-Class Questions (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@8c2dee8e62694eabbacd377ad4d3f171">
<h2 class="hd hd-2 unit-title">Lecture Video | Independence</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@video+block@90504792b17543409e4c71051da9e43b">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="video" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@video+block@90504792b17543409e4c71051da9e43b">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Independence</h3>
<div
id="video_90504792b17543409e4c71051da9e43b"
class="video closed"
data-metadata='{"streams": "1.00:1vQ2x5O_xqk", "autohideHtml5": false, "autoplay": false, "transcriptTranslationUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@90504792b17543409e4c71051da9e43b/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@90504792b17543409e4c71051da9e43b/handler/publish_completion", "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@90504792b17543409e4c71051da9e43b/handler/transcript/available_translations", "completionEnabled": false, "generalSpeed": 1.0, "ytTestTimeout": 1500, "ytMetadataEndpoint": "", "ytApiUrl": "https://www.youtube.com/iframe_api", "showCaptions": "true", "recordedYoutubeIsAvailable": true, "sources": ["https://archive.org/download/MIT6.042JS15/MIT6_042JS15_probindep_video_ipod.mp4"], "end": 0.0, "saveStateEnabled": false, "prioritizeHls": false, "savedVideoPosition": 0.0, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "start": 0.0, "saveStateUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@90504792b17543409e4c71051da9e43b/handler/xmodule_handler/save_user_state", "speed": null, "autoAdvance": false, "completionPercentage": 0.95, "duration": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="90504792b17543409e4c71051da9e43b"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_90504792b17543409e4c71051da9e43b">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_90504792b17543409e4c71051da9e43b">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://archive.org/download/MIT6.042JS15/MIT6_042JS15_probindep_video_ipod.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@90504792b17543409e4c71051da9e43b/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@90504792b17543409e4c71051da9e43b/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@8b09718c8b4b453f89ad599b31ba58ae">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="html" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@8b09718c8b4b453f89ad599b31ba58ae">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Download a copy of the slides for <a href="/assets/courseware/v1/fe44f539d627589c4453a69e219be9b9/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Independence.pdf" target="[object Object]">Independence (PDF)</a></p>
<p><a href="/assets/courseware/v1/ebdbf16069e8d81573ed7108825b475a/asset-v1:OCW+6.042J+2T2019+type@asset+block/Independence_4.3_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@12d34a3f43754f328d69567d6a061ae5">
<h2 class="hd hd-2 unit-title">Exercise | Independent Dice Rolls</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@592312f77d09488f89b82d19684ecbae">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="html" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@592312f77d09488f89b82d19684ecbae">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Consider the following events for rolling 2 dice:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi> <mi>i</mi> </msub></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mi></mi> <mo>:=</mo> <mtext>the first die is a </mtext> </math><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi> </math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">]</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"> <msub> <mi>B</mi> <mi>i</mi> </msub></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mi></mi> <mo>:=</mo> <mtext>the second die is a </mtext> </math><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi> </math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∈<!-- ∈ --></mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">]</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"> <msub> <mi>S</mi> <mi>i</mi> </msub></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi></mi><mo>:=</mo></math> sum of the dice is<mtext> </mtext> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> , for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∈<!-- ∈ --></mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>,</mo> <mn>12</mn> <mo stretchy="false">]</mo></math></p>
<p></p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@1a9c8b52086f497dbed270861850215e">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="problem" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="True" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@1a9c8b52086f497dbed270861850215e">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_1a9c8b52086f497dbed270861850215e" class="problems-wrapper" role="group"
aria-labelledby="1a9c8b52086f497dbed270861850215e-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@1a9c8b52086f497dbed270861850215e" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@1a9c8b52086f497dbed270861850215e/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="1a9c8b52086f497dbed270861850215e-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@1a9c8b52086f497dbed270861850215e-problem-progress" tabindex="-1">
Question
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@1a9c8b52086f497dbed270861850215e-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Which of the following events are independent?</p>
<div class="choicegroup capa_inputtype" id="inputtype_1a9c8b52086f497dbed270861850215e_2_1">
<fieldset aria-describedby="status_1a9c8b52086f497dbed270861850215e_2_1">
<div class="field">
<input type="checkbox" name="input_1a9c8b52086f497dbed270861850215e_2_1[]" id="input_1a9c8b52086f497dbed270861850215e_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="1a9c8b52086f497dbed270861850215e_2_1-choice_0-label" for="input_1a9c8b52086f497dbed270861850215e_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_1a9c8b52086f497dbed270861850215e_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>B</mi>
<mn>1</mn>
</msub>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_1a9c8b52086f497dbed270861850215e_2_1[]" id="input_1a9c8b52086f497dbed270861850215e_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="1a9c8b52086f497dbed270861850215e_2_1-choice_1-label" for="input_1a9c8b52086f497dbed270861850215e_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_1a9c8b52086f497dbed270861850215e_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</math>and <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>B</mi>
<mn>6</mn>
</msub>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_1a9c8b52086f497dbed270861850215e_2_1[]" id="input_1a9c8b52086f497dbed270861850215e_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="1a9c8b52086f497dbed270861850215e_2_1-choice_2-label" for="input_1a9c8b52086f497dbed270861850215e_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_1a9c8b52086f497dbed270861850215e_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>3</mn>
</msub>
</math>and <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>S</mi>
<mn>2</mn>
</msub>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_1a9c8b52086f497dbed270861850215e_2_1[]" id="input_1a9c8b52086f497dbed270861850215e_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="1a9c8b52086f497dbed270861850215e_2_1-choice_3-label" for="input_1a9c8b52086f497dbed270861850215e_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_1a9c8b52086f497dbed270861850215e_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</math>and <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>S</mi>
<mn>6</mn>
</msub>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_1a9c8b52086f497dbed270861850215e_2_1[]" id="input_1a9c8b52086f497dbed270861850215e_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="1a9c8b52086f497dbed270861850215e_2_1-choice_4-label" for="input_1a9c8b52086f497dbed270861850215e_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_1a9c8b52086f497dbed270861850215e_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</math>and <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>S</mi>
<mn>7</mn>
</msub>
</math>
</label>
</div>
<span id="answer_1a9c8b52086f497dbed270861850215e_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_1a9c8b52086f497dbed270861850215e_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_1a9c8b52086f497dbed270861850215e_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Question" />
<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_1a9c8b52086f497dbed270861850215e" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_1a9c8b52086f497dbed270861850215e">
<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="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="1a9c8b52086f497dbed270861850215e-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show 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="1a9c8b52086f497dbed270861850215e-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="1a9c8b52086f497dbed270861850215e-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="1a9c8b52086f497dbed270861850215e-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>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@02398c2fc98740fcaa7fcbd977fe88f7">
<h2 class="hd hd-2 unit-title">Lecture Video | Mutual Independence</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@video+block@91fa76b32da748fc95f0fd9576e12ea2">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="video" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@video+block@91fa76b32da748fc95f0fd9576e12ea2">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Mutual Independence</h3>
<div
id="video_91fa76b32da748fc95f0fd9576e12ea2"
class="video closed"
data-metadata='{"streams": "1.00:wJzBU7Do1ls", "autohideHtml5": false, "autoplay": false, "transcriptTranslationUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@91fa76b32da748fc95f0fd9576e12ea2/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@91fa76b32da748fc95f0fd9576e12ea2/handler/publish_completion", "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@91fa76b32da748fc95f0fd9576e12ea2/handler/transcript/available_translations", "completionEnabled": false, "generalSpeed": 1.0, "ytTestTimeout": 1500, "ytMetadataEndpoint": "", "ytApiUrl": "https://www.youtube.com/iframe_api", "showCaptions": "true", "recordedYoutubeIsAvailable": true, "sources": ["https://archive.org/download/MIT6.042JS15/MIT6_042JS15_probmutual_video_ipod.mp4"], "end": 0.0, "saveStateEnabled": false, "prioritizeHls": false, "savedVideoPosition": 0.0, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "start": 0.0, "saveStateUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@91fa76b32da748fc95f0fd9576e12ea2/handler/xmodule_handler/save_user_state", "speed": null, "autoAdvance": false, "completionPercentage": 0.95, "duration": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="91fa76b32da748fc95f0fd9576e12ea2"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_91fa76b32da748fc95f0fd9576e12ea2">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_91fa76b32da748fc95f0fd9576e12ea2">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://archive.org/download/MIT6.042JS15/MIT6_042JS15_probmutual_video_ipod.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@91fa76b32da748fc95f0fd9576e12ea2/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@91fa76b32da748fc95f0fd9576e12ea2/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@9e702eb5177644a882930da1c88c9046">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="html" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@9e702eb5177644a882930da1c88c9046">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Download a copy of the slides for <a href="/assets/courseware/v1/6c49d7f0ff87d2c57d83fa8e01d4ac3d/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_MutualIndepend.pdf" target="[object Object]">Mutual Independence (PDF)</a></p>
<p><a href="/assets/courseware/v1/f0d8bca1da4442d9dfe505508b136baf/asset-v1:OCW+6.042J+2T2019+type@asset+block/Mutual_Independence_4.3_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@4be9a325e54e44e89bdaf96bc95c0139">
<h2 class="hd hd-2 unit-title">Exercise | Mutually Independent Dice Rolls</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@75551b086c56438d9b088b80ecae60ac">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="html" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@75551b086c56438d9b088b80ecae60ac">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Consider the following events the rolling 3 dice:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi> <mi>i</mi> </msub></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mi></mi> <mo>:=</mo> <mtext>the first die is a </mtext> </math><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi> </math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">]</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"> <msub> <mi>B</mi> <mi>i</mi> </msub></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mi></mi> <mo>:=</mo> <mtext>the second die is a </mtext> </math><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi> </math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∈<!-- ∈ --></mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">]</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi></msub></math> <mi>i</mi> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi></mi><mo>:=</mo></math> <mtext>the third die is a </mtext> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> , for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∈<!-- ∈ --></mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">]</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi></msub></math> <mi>i</mi> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi></mi><mo>:=</mo></math> sum of the<mtext> dice is</mtext> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> , for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∈<!-- ∈ --></mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>,</mo> <mn>18</mn> <mo stretchy="false">]</mo></math></p>
<p></p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@8a47e4021a474b4484a514022893a5ca">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="problem" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="True" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@8a47e4021a474b4484a514022893a5ca">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_8a47e4021a474b4484a514022893a5ca" class="problems-wrapper" role="group"
aria-labelledby="8a47e4021a474b4484a514022893a5ca-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@8a47e4021a474b4484a514022893a5ca" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@8a47e4021a474b4484a514022893a5ca/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="8a47e4021a474b4484a514022893a5ca-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@8a47e4021a474b4484a514022893a5ca-problem-progress" tabindex="-1">
Question
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@8a47e4021a474b4484a514022893a5ca-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p> Which of the following events are mutually independent?</p>
<div class="choicegroup capa_inputtype" id="inputtype_8a47e4021a474b4484a514022893a5ca_2_1">
<fieldset aria-describedby="status_8a47e4021a474b4484a514022893a5ca_2_1">
<div class="field">
<input type="checkbox" name="input_8a47e4021a474b4484a514022893a5ca_2_1[]" id="input_8a47e4021a474b4484a514022893a5ca_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="8a47e4021a474b4484a514022893a5ca_2_1-choice_0-label" for="input_8a47e4021a474b4484a514022893a5ca_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_8a47e4021a474b4484a514022893a5ca_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>B</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>C</mi>
<mn>1</mn>
</msub>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_8a47e4021a474b4484a514022893a5ca_2_1[]" id="input_8a47e4021a474b4484a514022893a5ca_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="8a47e4021a474b4484a514022893a5ca_2_1-choice_1-label" for="input_8a47e4021a474b4484a514022893a5ca_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8a47e4021a474b4484a514022893a5ca_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>B</mi>
<mn>6</mn>
</msub>
<mo>,</mo>
<msub>
<mi>C</mi>
<mn>3</mn>
</msub>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_8a47e4021a474b4484a514022893a5ca_2_1[]" id="input_8a47e4021a474b4484a514022893a5ca_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="8a47e4021a474b4484a514022893a5ca_2_1-choice_2-label" for="input_8a47e4021a474b4484a514022893a5ca_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_8a47e4021a474b4484a514022893a5ca_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>B</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>S</mi>
<mn>3</mn>
</msub>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_8a47e4021a474b4484a514022893a5ca_2_1[]" id="input_8a47e4021a474b4484a514022893a5ca_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="8a47e4021a474b4484a514022893a5ca_2_1-choice_3-label" for="input_8a47e4021a474b4484a514022893a5ca_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_8a47e4021a474b4484a514022893a5ca_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>B</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>S</mi>
<mn>7</mn>
</msub>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_8a47e4021a474b4484a514022893a5ca_2_1[]" id="input_8a47e4021a474b4484a514022893a5ca_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="8a47e4021a474b4484a514022893a5ca_2_1-choice_4-label" for="input_8a47e4021a474b4484a514022893a5ca_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_8a47e4021a474b4484a514022893a5ca_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>B</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>S</mi>
<mn>8</mn>
</msub>
</math>
</label>
</div>
<span id="answer_8a47e4021a474b4484a514022893a5ca_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_8a47e4021a474b4484a514022893a5ca_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_8a47e4021a474b4484a514022893a5ca_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Question" />
<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_8a47e4021a474b4484a514022893a5ca" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_8a47e4021a474b4484a514022893a5ca">
<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="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="8a47e4021a474b4484a514022893a5ca-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show 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="8a47e4021a474b4484a514022893a5ca-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="8a47e4021a474b4484a514022893a5ca-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="8a47e4021a474b4484a514022893a5ca-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>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@969c6901960347779daf1f3e10604a24">
<h2 class="hd hd-2 unit-title">Exercise | Independent vs Disjoint</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@323fab3d7c9a42f399cfb1fa1661afd4">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="problem" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="True" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@323fab3d7c9a42f399cfb1fa1661afd4">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_323fab3d7c9a42f399cfb1fa1661afd4" class="problems-wrapper" role="group"
aria-labelledby="323fab3d7c9a42f399cfb1fa1661afd4-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@323fab3d7c9a42f399cfb1fa1661afd4" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@323fab3d7c9a42f399cfb1fa1661afd4/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="323fab3d7c9a42f399cfb1fa1661afd4-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@323fab3d7c9a42f399cfb1fa1661afd4-problem-progress" tabindex="-1">
Independent vs Disjoint
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@323fab3d7c9a42f399cfb1fa1661afd4-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p> Can two events be both disjoint and independent? </p>
<div class="choicegroup capa_inputtype" id="inputtype_323fab3d7c9a42f399cfb1fa1661afd4_2_1">
<fieldset aria-describedby="status_323fab3d7c9a42f399cfb1fa1661afd4_2_1">
<div class="field">
<input type="checkbox" name="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1[]" id="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="323fab3d7c9a42f399cfb1fa1661afd4_2_1-choice_0-label" for="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_323fab3d7c9a42f399cfb1fa1661afd4_2_1"> Never
</label>
</div>
<div class="field">
<input type="checkbox" name="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1[]" id="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="323fab3d7c9a42f399cfb1fa1661afd4_2_1-choice_1-label" for="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_323fab3d7c9a42f399cfb1fa1661afd4_2_1"> Only if the two terms are equivalent
</label>
</div>
<div class="field">
<input type="checkbox" name="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1[]" id="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="323fab3d7c9a42f399cfb1fa1661afd4_2_1-choice_2-label" for="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_323fab3d7c9a42f399cfb1fa1661afd4_2_1"> Only if the disjoint events are always independent
</label>
</div>
<div class="field">
<input type="checkbox" name="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1[]" id="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="323fab3d7c9a42f399cfb1fa1661afd4_2_1-choice_3-label" for="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_323fab3d7c9a42f399cfb1fa1661afd4_2_1"> Only if at least one event is empty
</label>
</div>
<div class="field">
<input type="checkbox" name="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1[]" id="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="323fab3d7c9a42f399cfb1fa1661afd4_2_1-choice_4-label" for="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_323fab3d7c9a42f399cfb1fa1661afd4_2_1"> Only if both events have are empty
</label>
</div>
<div class="field">
<input type="checkbox" name="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1[]" id="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="323fab3d7c9a42f399cfb1fa1661afd4_2_1-choice_5-label" for="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_323fab3d7c9a42f399cfb1fa1661afd4_2_1"> Only if at least one event has probability 1 or is empty
</label>
</div>
<div class="field">
<input type="checkbox" name="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1[]" id="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="323fab3d7c9a42f399cfb1fa1661afd4_2_1-choice_6-label" for="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_323fab3d7c9a42f399cfb1fa1661afd4_2_1"> Only if both events have probability 1 or is empty
</label>
</div>
<div class="field">
<input type="checkbox" name="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1[]" id="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="323fab3d7c9a42f399cfb1fa1661afd4_2_1-choice_7-label" for="input_323fab3d7c9a42f399cfb1fa1661afd4_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_323fab3d7c9a42f399cfb1fa1661afd4_2_1"> Always
</label>
</div>
<span id="answer_323fab3d7c9a42f399cfb1fa1661afd4_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_323fab3d7c9a42f399cfb1fa1661afd4_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_323fab3d7c9a42f399cfb1fa1661afd4_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Independent vs Disjoint" />
<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_323fab3d7c9a42f399cfb1fa1661afd4" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_323fab3d7c9a42f399cfb1fa1661afd4">
<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="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="323fab3d7c9a42f399cfb1fa1661afd4-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show 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="323fab3d7c9a42f399cfb1fa1661afd4-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="323fab3d7c9a42f399cfb1fa1661afd4-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="323fab3d7c9a42f399cfb1fa1661afd4-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>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@a8c01b46ec60476dbda3993655e85104">
<h2 class="hd hd-2 unit-title">Exercise | Labeled Balls</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@92fde45096964864a63f7a7e10b67e03">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="problem" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="True" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@92fde45096964864a63f7a7e10b67e03">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_92fde45096964864a63f7a7e10b67e03" class="problems-wrapper" role="group"
aria-labelledby="92fde45096964864a63f7a7e10b67e03-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@92fde45096964864a63f7a7e10b67e03" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@92fde45096964864a63f7a7e10b67e03/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="92fde45096964864a63f7a7e10b67e03-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@92fde45096964864a63f7a7e10b67e03-problem-progress" tabindex="-1">
Labeled Balls
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@92fde45096964864a63f7a7e10b67e03-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_92fde45096964864a63f7a7e10b67e03_2_1">
<fieldset aria-describedby="status_92fde45096964864a63f7a7e10b67e03_2_1">
<legend id="92fde45096964864a63f7a7e10b67e03_2_1-legend" class="response-fieldset-legend field-group-hd">I have four balls in an urn, each one distinctly labeled with either 110, 101, 011, or 000. I draw one at random.
<br/>
<br/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</math> denote the event that I draw a ball with a 1 in the <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
<mi>t</mi>
</mrow>
</msup>
</math> position. Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>3</mn>
</msub>
</math> be defined similarly.
<br/>
<br/>
Which of the following are true about <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<msub>
<mi>A</mi>
<mn>3</mn>
</msub>
</math> ?</legend>
<div class="field">
<input type="checkbox" name="input_92fde45096964864a63f7a7e10b67e03_2_1[]" id="input_92fde45096964864a63f7a7e10b67e03_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="92fde45096964864a63f7a7e10b67e03_2_1-choice_0-label" for="input_92fde45096964864a63f7a7e10b67e03_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_92fde45096964864a63f7a7e10b67e03_2_1"> They are mutually independent
</label>
</div>
<div class="field">
<input type="checkbox" name="input_92fde45096964864a63f7a7e10b67e03_2_1[]" id="input_92fde45096964864a63f7a7e10b67e03_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="92fde45096964864a63f7a7e10b67e03_2_1-choice_1-label" for="input_92fde45096964864a63f7a7e10b67e03_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_92fde45096964864a63f7a7e10b67e03_2_1"> They are pairwise independent
</label>
</div>
<div class="field">
<input type="checkbox" name="input_92fde45096964864a63f7a7e10b67e03_2_1[]" id="input_92fde45096964864a63f7a7e10b67e03_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="92fde45096964864a63f7a7e10b67e03_2_1-choice_2-label" for="input_92fde45096964864a63f7a7e10b67e03_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_92fde45096964864a63f7a7e10b67e03_2_1"> They are pairwise disjoint
</label>
</div>
<div class="field">
<input type="checkbox" name="input_92fde45096964864a63f7a7e10b67e03_2_1[]" id="input_92fde45096964864a63f7a7e10b67e03_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="92fde45096964864a63f7a7e10b67e03_2_1-choice_3-label" for="input_92fde45096964864a63f7a7e10b67e03_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_92fde45096964864a63f7a7e10b67e03_2_1"> They all have equal probability
</label>
</div>
<span id="answer_92fde45096964864a63f7a7e10b67e03_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_92fde45096964864a63f7a7e10b67e03_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_92fde45096964864a63f7a7e10b67e03_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Labeled Balls" />
<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_92fde45096964864a63f7a7e10b67e03" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_92fde45096964864a63f7a7e10b67e03">
<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="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="92fde45096964864a63f7a7e10b67e03-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show 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="92fde45096964864a63f7a7e10b67e03-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="92fde45096964864a63f7a7e10b67e03-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="92fde45096964864a63f7a7e10b67e03-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>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@1bd83f3c4876480d9413df534d6cb062">
<h2 class="hd hd-2 unit-title">Exercise | Paradox</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@0708c715c3b9497b8c4935fed636d934">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="21a74714e12211ef932016ffeba94eb7" data-block-type="problem" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="True" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@0708c715c3b9497b8c4935fed636d934">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_0708c715c3b9497b8c4935fed636d934" class="problems-wrapper" role="group"
aria-labelledby="0708c715c3b9497b8c4935fed636d934-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@0708c715c3b9497b8c4935fed636d934" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@0708c715c3b9497b8c4935fed636d934/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="0708c715c3b9497b8c4935fed636d934-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@0708c715c3b9497b8c4935fed636d934-problem-progress" tabindex="-1">
Paradox
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@0708c715c3b9497b8c4935fed636d934-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_0708c715c3b9497b8c4935fed636d934_2_1">
<fieldset aria-describedby="status_0708c715c3b9497b8c4935fed636d934_2_1">
<legend id="0708c715c3b9497b8c4935fed636d934_2_1-legend" class="response-fieldset-legend field-group-hd">Suppose event A is evidence that event H occurred, and event B is also evidence that H occurred. Is (A or B) necessarily evidence that H occurred? A precise way of thinking about evidence is that E is evidence for H if <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>E</mi>
<mo stretchy="false">]</mo>
<mo>&gt;</mo>
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>H</mi>
<mo stretchy="false">]</mo>
</math>.</legend>
<div class="field">
<input type="radio" name="input_0708c715c3b9497b8c4935fed636d934_2_1" id="input_0708c715c3b9497b8c4935fed636d934_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="0708c715c3b9497b8c4935fed636d934_2_1-choice_0-label" for="input_0708c715c3b9497b8c4935fed636d934_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_0708c715c3b9497b8c4935fed636d934_2_1"> Yes
</label>
</div>
<div class="field">
<input type="radio" name="input_0708c715c3b9497b8c4935fed636d934_2_1" id="input_0708c715c3b9497b8c4935fed636d934_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="0708c715c3b9497b8c4935fed636d934_2_1-choice_1-label" for="input_0708c715c3b9497b8c4935fed636d934_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_0708c715c3b9497b8c4935fed636d934_2_1"> No
</label>
</div>
<span id="answer_0708c715c3b9497b8c4935fed636d934_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_0708c715c3b9497b8c4935fed636d934_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_0708c715c3b9497b8c4935fed636d934_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Paradox" />
<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_0708c715c3b9497b8c4935fed636d934" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_0708c715c3b9497b8c4935fed636d934">
<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="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="0708c715c3b9497b8c4935fed636d934-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show 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="0708c715c3b9497b8c4935fed636d934-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="0708c715c3b9497b8c4935fed636d934-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="0708c715c3b9497b8c4935fed636d934-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>