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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/cc1af0d86db21d2bb9c32e7e91e08fe9/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session31.pdf" target="[object Object]">Chapter 18.1–18.3 (PDF)</a> of <em>Mathematics for Computer Science</em> for 4.4 Random Variables, Density Functions.</p>
<p>View the <a href="/assets/courseware/v1/7f7138d729b6c3b8709acf4e52289c82/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp31.pdf" target="[object Object]">Section 4.4 In-Class Questions (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Bigger Number Game</h2>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/1e6408719ddba950279d486c964d0a8d/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_BigerNmberGme.pdf" target="[object Object]">Bigger Number Game (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Random Variables: Independence</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Odd Heads and Matches</h2>
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<p>We flip 3 coins. Let</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mi>C</mi> <mo>:=</mo> <mi mathvariant="normal">#<!-- # --></mi> <mtext>heads</mtext> </math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mi>M</mi> <mo>:=</mo> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mtext>if all flips match</mtext> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mtext>otherwise</mtext> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true"></mo> </mrow> </math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mi>O</mi> <mo>:=</mo> <mtext> odd #heads</mtext> </math></p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"> <msub> <mi>I</mi> <mi>O</mi> </msub> </math> be the indicator variable for <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>O</mi> </math>.</p>
<p>We want to show that <math xmlns="http://www.w3.org/1998/Math/MathML"> <msub> <mi>I</mi> <mi>O</mi> </msub> </math> and <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>M</mi> </math> are independent.</p>
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1. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<msub>
<mi>I</mi>
<mi>O</mi>
</msub>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
</math></label>
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<span class="trailing_text" id="trailing_text_7c355f26a8634f249d9b60ee630363cf_2_1"/>
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\(\)
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<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
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<mo>=</mo>
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<mo stretchy="false">]</mo>
</math></label>
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<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
</math></label>
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<label class="problem-group-label" for="input_7c355f26a8634f249d9b60ee630363cf_5_1" id="label_7c355f26a8634f249d9b60ee630363cf_5_1">4. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo>=</mo>
<mn>1</mn>
<mtext>&#160;AND&#160;</mtext>
<msub>
<mi>I</mi>
<mi>O</mi>
</msub>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
</math></label>
<input type="text" name="input_7c355f26a8634f249d9b60ee630363cf_5_1" id="input_7c355f26a8634f249d9b60ee630363cf_5_1" data-input-id="7c355f26a8634f249d9b60ee630363cf_5_1" value="" aria-describedby="status_7c355f26a8634f249d9b60ee630363cf_5_1" size="20"/>
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<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo>=</mo>
<mn>0</mn>
<mtext>&#160;AND&#160;</mtext>
<msub>
<mi>I</mi>
<mi>O</mi>
</msub>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
</math></label>
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<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo>=</mo>
<mn>1</mn>
<mtext>&#160;AND&#160;</mtext>
<msub>
<mi>I</mi>
<mi>O</mi>
</msub>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">]</mo>
</math></label>
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<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>M</mi>
<mo>=</mo>
<mn>0</mn>
<mtext>&#160;AND&#160;</mtext>
<msub>
<mi>I</mi>
<mi>O</mi>
</msub>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">]</mo>
</math></label>
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<h2 class="hd hd-2 unit-title">Lecture Video | Random Variables: Uniform & Binomial</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Late for a Date</h2>
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<ul>
<li style="margin: 0px 0px 4px; padding: 0px; border: 0px; font-size: 1em; color: #333333;">Sean and Jess have reservations at Trident at 10am. Sean is walking from Back Bay and Jess is taking the T from Central Square.</li>
</ul>
</ul>
<ul>
<ul>
<li style="margin: 0px 0px 4px; padding: 0px; border: 0px; font-size: 1em; color: #333333;">The T is unreliable, so Jess will arrive at some time between 9:50am and 10:10am (whole minutes only) with uniform probability.</li>
</ul>
</ul>
<ul>
<ul>
<li style="margin: 0px 0px 4px; padding: 0px; border: 0px; font-size: 1em; color: #333333;">On Sean's way to Trident, there are 10 stores that he likes. He will spend 1 minute in a given store independently with probability <span class="MathJax_Preview" style="margin: 0px; padding: 0px; border: 0px; color: inherit;"></span><span class="MathJax_SVG" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math>" role="presentation" style="margin: 0px; padding: 0px; border: 0px; display: inline-block; line-height: normal; font-size: 12px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.658ex" height="3.699ex" viewbox="0 -1038.1 713.9 1592.7" role="img" focusable="false" style="vertical-align: -1.288ex;" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="473" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="84" y="622"><path stroke-width="1" id="MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></use><use transform="scale(0.707)" xlink:href="#MJMAIN-33" x="84" y="-581"><path stroke-width="1" id="MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation" style="margin: 0px; padding: 1px 0px 0px !important; border: 0px !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); user-select: none; position: absolute !important; height: 1px !important; width: 1px !important; overflow: hidden !important; display: block !important; transition: none 0s ease 0s;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math></span></span>. If he doesn't visit any of the stores, he will arrive at Trident at 9:55am (if he goes to all of them, he will be 5 minutes late).</li>
</ul>
</ul>
<ul>
<ul>
<li style="margin: 0px 0px 4px; padding: 0px; border: 0px; font-size: 1em; color: #333333;">Their arrival times are independent.</li>
</ul>
</ul>
</div>
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<p><em> Please answer in fractions of the form x/y. </em></p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_951e4e1695114035b61d2eb6dd5b0e49_2_1" class=" capa_inputtype textline">
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<label class="problem-group-label" for="input_951e4e1695114035b61d2eb6dd5b0e49_2_1" id="label_951e4e1695114035b61d2eb6dd5b0e49_2_1">1. Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>J</mi></math> be the random variable for Jess' minute of arrival. I.e. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>J</mi> </math> gives values in <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">[</mo>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>10</mn>
<mo>,</mo>
<mn>10</mn>
<mo stretchy="false">]</mo>
</math>. What is the probability that Jess will be exactly on time (i.e.<math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>J</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">]</mo>
</math> )?</label>
<input type="text" name="input_951e4e1695114035b61d2eb6dd5b0e49_2_1" id="input_951e4e1695114035b61d2eb6dd5b0e49_2_1" aria-describedby="status_951e4e1695114035b61d2eb6dd5b0e49_2_1" value="" size="20"/>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_951e4e1695114035b61d2eb6dd5b0e49_3_1" class=" capa_inputtype textline">
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<label class="problem-group-label" for="input_951e4e1695114035b61d2eb6dd5b0e49_3_1" id="label_951e4e1695114035b61d2eb6dd5b0e49_3_1">2. Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
</math> be the random variable for Sean's minute of arrival. I.e. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
</math> gives values in <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">[</mo>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>5</mn>
<mo>,</mo>
<mn>5</mn>
<mo stretchy="false">]</mo>
</math>. What is the probability that Sean will be exactly on time (i.e.<math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>S</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">]</mo>
</math> )?</label>
<input type="text" name="input_951e4e1695114035b61d2eb6dd5b0e49_3_1" id="input_951e4e1695114035b61d2eb6dd5b0e49_3_1" aria-describedby="status_951e4e1695114035b61d2eb6dd5b0e49_3_1" value="" size="20"/>
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<label class="problem-group-label" for="input_951e4e1695114035b61d2eb6dd5b0e49_4_1" id="label_951e4e1695114035b61d2eb6dd5b0e49_4_1">3. What is the probability that they will arrive at the same time?</label>
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<h2 class="hd hd-2 unit-title">Exercise | A Random Number</h2>
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<p>Here is a process to construct a random number:</p>
<ol>
<li>Flip a biased coin that comes up heads with probability <math xmlns="http://www.w3.org/1998/Math/MathML"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </math></math>.</li>
<li> * If you get heads, you roll a fair die and return the result. </li>
</ol>
<p> * Otherwise, you flip a fair coin 3 times and return twice the number of heads. </p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>N</mi> </math> be the number that you return.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math> be the indicator random variable for the first coin flip (1 if heads and 0 if tails).</p>
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Questions
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p><em> Please answer in the form of a decimal with three significant figures.</em></p>
<div id="formulaequationinput_76bad1864f6c490e807bd8e6664c89d9_2_1" class="inputtype formulaequationinput">
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<label class="problem-group-label" for="input_76bad1864f6c490e807bd8e6664c89d9_2_1" id="label_76bad1864f6c490e807bd8e6664c89d9_2_1">1. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>N</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">]</mo>
</math>?</label>
<input type="text" name="input_76bad1864f6c490e807bd8e6664c89d9_2_1" id="input_76bad1864f6c490e807bd8e6664c89d9_2_1" data-input-id="76bad1864f6c490e807bd8e6664c89d9_2_1" value="" aria-describedby="status_76bad1864f6c490e807bd8e6664c89d9_2_1" size="20"/>
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<span class="status unanswered" id="status_76bad1864f6c490e807bd8e6664c89d9_2_1" data-tooltip="Not yet answered.">
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<div id="input_76bad1864f6c490e807bd8e6664c89d9_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
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<span id="solution_76bad1864f6c490e807bd8e6664c89d9_solution_1"/>
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<label class="problem-group-label" for="input_76bad1864f6c490e807bd8e6664c89d9_3_1" id="label_76bad1864f6c490e807bd8e6664c89d9_3_1">2. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>N</mi>
<mo>=</mo>
<mn>3</mn>
<mo stretchy="false">]</mo>
</math>?</label>
<input type="text" name="input_76bad1864f6c490e807bd8e6664c89d9_3_1" id="input_76bad1864f6c490e807bd8e6664c89d9_3_1" data-input-id="76bad1864f6c490e807bd8e6664c89d9_3_1" value="" aria-describedby="status_76bad1864f6c490e807bd8e6664c89d9_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_76bad1864f6c490e807bd8e6664c89d9_3_1"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_76bad1864f6c490e807bd8e6664c89d9_3_1" class="answer"/>
<div id="input_76bad1864f6c490e807bd8e6664c89d9_3_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="solution-span">
<span id="solution_76bad1864f6c490e807bd8e6664c89d9_solution_2"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><br/>
<div id="formulaequationinput_76bad1864f6c490e807bd8e6664c89d9_4_1" class="inputtype formulaequationinput">
<div class="unanswered">
<label class="problem-group-label" for="input_76bad1864f6c490e807bd8e6664c89d9_4_1" id="label_76bad1864f6c490e807bd8e6664c89d9_4_1">3. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>N</mi>
<mo>=</mo>
<mn>6</mn>
<mo stretchy="false">]</mo>
</math>?</label>
<input type="text" name="input_76bad1864f6c490e807bd8e6664c89d9_4_1" id="input_76bad1864f6c490e807bd8e6664c89d9_4_1" data-input-id="76bad1864f6c490e807bd8e6664c89d9_4_1" value="" aria-describedby="status_76bad1864f6c490e807bd8e6664c89d9_4_1" size="20"/>
<span class="trailing_text" id="trailing_text_76bad1864f6c490e807bd8e6664c89d9_4_1"/>
<span class="status unanswered" id="status_76bad1864f6c490e807bd8e6664c89d9_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_76bad1864f6c490e807bd8e6664c89d9_4_1" class="answer"/>
<div id="input_76bad1864f6c490e807bd8e6664c89d9_4_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="solution-span">
<span id="solution_76bad1864f6c490e807bd8e6664c89d9_solution_3"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><br/>
<div id="formulaequationinput_76bad1864f6c490e807bd8e6664c89d9_5_1" class="inputtype formulaequationinput">
<div class="unanswered">
<label class="problem-group-label" for="input_76bad1864f6c490e807bd8e6664c89d9_5_1" id="label_76bad1864f6c490e807bd8e6664c89d9_5_1">4. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>N</mi>
<mo>=</mo>
<mn>7</mn>
<mo stretchy="false">]</mo>
</math>?</label>
<input type="text" name="input_76bad1864f6c490e807bd8e6664c89d9_5_1" id="input_76bad1864f6c490e807bd8e6664c89d9_5_1" data-input-id="76bad1864f6c490e807bd8e6664c89d9_5_1" value="" aria-describedby="status_76bad1864f6c490e807bd8e6664c89d9_5_1" size="20"/>
<span class="trailing_text" id="trailing_text_76bad1864f6c490e807bd8e6664c89d9_5_1"/>
<span class="status unanswered" id="status_76bad1864f6c490e807bd8e6664c89d9_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_76bad1864f6c490e807bd8e6664c89d9_5_1" class="answer"/>
<div id="input_76bad1864f6c490e807bd8e6664c89d9_5_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="solution-span">
<span id="solution_76bad1864f6c490e807bd8e6664c89d9_solution_4"/>
</div><br/>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div id="formulaequationinput_76bad1864f6c490e807bd8e6664c89d9_6_1" class="inputtype formulaequationinput">
<div class="unanswered">
<label class="problem-group-label" for="input_76bad1864f6c490e807bd8e6664c89d9_6_1" id="label_76bad1864f6c490e807bd8e6664c89d9_6_1">5. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>N</mi>
<mo>=</mo>
<mn>6</mn>
<mspace width="thickmathspace"/>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mspace width="thickmathspace"/>
<mi>F</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">]</mo>
</math>?</label>
<input type="text" name="input_76bad1864f6c490e807bd8e6664c89d9_6_1" id="input_76bad1864f6c490e807bd8e6664c89d9_6_1" data-input-id="76bad1864f6c490e807bd8e6664c89d9_6_1" value="" aria-describedby="status_76bad1864f6c490e807bd8e6664c89d9_6_1" size="20"/>
<span class="trailing_text" id="trailing_text_76bad1864f6c490e807bd8e6664c89d9_6_1"/>
<span class="status unanswered" id="status_76bad1864f6c490e807bd8e6664c89d9_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_76bad1864f6c490e807bd8e6664c89d9_6_1" class="answer"/>
<div id="input_76bad1864f6c490e807bd8e6664c89d9_6_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div id="formulaequationinput_76bad1864f6c490e807bd8e6664c89d9_7_1" class="inputtype formulaequationinput">
<div class="unanswered">
<label class="problem-group-label" for="input_76bad1864f6c490e807bd8e6664c89d9_7_1" id="label_76bad1864f6c490e807bd8e6664c89d9_7_1">6. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>F</mi>
<mo>=</mo>
<mn>0</mn>
<mspace width="thickmathspace"/>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mspace width="thickmathspace"/>
<mi>N</mi>
<mo>=</mo>
<mn>6</mn>
<mo stretchy="false">]</mo>
</math>?</label>
<input type="text" name="input_76bad1864f6c490e807bd8e6664c89d9_7_1" id="input_76bad1864f6c490e807bd8e6664c89d9_7_1" data-input-id="76bad1864f6c490e807bd8e6664c89d9_7_1" value="" aria-describedby="status_76bad1864f6c490e807bd8e6664c89d9_7_1" size="20"/>
<span class="trailing_text" id="trailing_text_76bad1864f6c490e807bd8e6664c89d9_7_1"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div id="input_76bad1864f6c490e807bd8e6664c89d9_7_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="solution-span">
<span id="solution_76bad1864f6c490e807bd8e6664c89d9_solution_5"/>
</div><br/>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 7" role="group"><div id="formulaequationinput_76bad1864f6c490e807bd8e6664c89d9_8_1" class="inputtype formulaequationinput">
<div class="unanswered">
<label class="problem-group-label" for="input_76bad1864f6c490e807bd8e6664c89d9_8_1" id="label_76bad1864f6c490e807bd8e6664c89d9_8_1">7. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>N</mi>
<mo>+</mo>
<mi>F</mi>
<mo>=</mo>
<mn>5</mn>
<mo stretchy="false">]</mo>
</math>?</label>
<input type="text" name="input_76bad1864f6c490e807bd8e6664c89d9_8_1" id="input_76bad1864f6c490e807bd8e6664c89d9_8_1" data-input-id="76bad1864f6c490e807bd8e6664c89d9_8_1" value="" aria-describedby="status_76bad1864f6c490e807bd8e6664c89d9_8_1" size="20"/>
<span class="trailing_text" id="trailing_text_76bad1864f6c490e807bd8e6664c89d9_8_1"/>
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</span>
<p id="answer_76bad1864f6c490e807bd8e6664c89d9_8_1" class="answer"/>
<div id="input_76bad1864f6c490e807bd8e6664c89d9_8_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="solution-span">
<span id="solution_76bad1864f6c490e807bd8e6664c89d9_solution_6"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 8" role="group"><br/>
<div id="formulaequationinput_76bad1864f6c490e807bd8e6664c89d9_9_1" class="inputtype formulaequationinput">
<div class="unanswered">
<label class="problem-group-label" for="input_76bad1864f6c490e807bd8e6664c89d9_9_1" id="label_76bad1864f6c490e807bd8e6664c89d9_9_1">8. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>N</mi>
<mo>+</mo>
<mi>F</mi>
<mo>=</mo>
<mn>6</mn>
<mo stretchy="false">]</mo>
</math>?</label>
<input type="text" name="input_76bad1864f6c490e807bd8e6664c89d9_9_1" id="input_76bad1864f6c490e807bd8e6664c89d9_9_1" data-input-id="76bad1864f6c490e807bd8e6664c89d9_9_1" value="" aria-describedby="status_76bad1864f6c490e807bd8e6664c89d9_9_1" size="20"/>
<span class="trailing_text" id="trailing_text_76bad1864f6c490e807bd8e6664c89d9_9_1"/>
<span class="status unanswered" id="status_76bad1864f6c490e807bd8e6664c89d9_9_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_76bad1864f6c490e807bd8e6664c89d9_9_1" class="answer"/>
<div id="input_76bad1864f6c490e807bd8e6664c89d9_9_1_preview" class="equation">
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<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
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<h2 class="hd hd-2 unit-title">Exercise | PDF to CDF</h2>
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<label class="problem-group-label" for="input_f450ecdc6baa4dceaf56e8484feffe73_2_1" id="label_f450ecdc6baa4dceaf56e8484feffe73_2_1">Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>X</mi>
</math> be a uniformly distributed random variable on the interval [1,12]. What is the value of the Cumulative Distribution Function (CDF) at 8?
<br/>
<br/>
<em> Please enter your answer as a decimal with two significant figures.</em></label>
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<h2 class="hd hd-2 unit-title">Exercise | Dice and Coin Game</h2>
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Dice and Coin Game
</h3>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_a71b5b2c4f0a4ae4af4b0bf5227f0cf2_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_a71b5b2c4f0a4ae4af4b0bf5227f0cf2_2_1" id="label_a71b5b2c4f0a4ae4af4b0bf5227f0cf2_2_1">A fair die is tossed and its outcome is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>X</mi>
</math>. After that, we toss an independent fair coin <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>X</mi>
</math> times and count the number of heads. Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>Y</mi>
</math>denote the number of heads.
<br/>
<br/>
<em> Please answer as a fraciton in the form of x/y.</em>
<br/>
<br/>
1. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>Y</mi>
<mo>=</mo>
<mn>4</mn>
<mo stretchy="false">]</mo>
</math>?</label>
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<span class="trailing_text" id="trailing_text_a71b5b2c4f0a4ae4af4b0bf5227f0cf2_2_1"/>
<span class="status unanswered" id="status_a71b5b2c4f0a4ae4af4b0bf5227f0cf2_2_1" data-tooltip="Not yet answered.">
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_a71b5b2c4f0a4ae4af4b0bf5227f0cf2_3_1" class=" capa_inputtype textline">
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<label class="problem-group-label" for="input_a71b5b2c4f0a4ae4af4b0bf5227f0cf2_3_1" id="label_a71b5b2c4f0a4ae4af4b0bf5227f0cf2_3_1">2. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>=</mo>
<mn>5</mn>
<mspace width="thickmathspace"/>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mspace width="thickmathspace"/>
<mi>Y</mi>
<mo>=</mo>
<mn>4</mn>
<mo stretchy="false">]</mo>
</math>?</label>
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<span class="trailing_text" id="trailing_text_a71b5b2c4f0a4ae4af4b0bf5227f0cf2_3_1"/>
<span class="status unanswered" id="status_a71b5b2c4f0a4ae4af4b0bf5227f0cf2_3_1" data-tooltip="Not yet answered.">
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