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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/e56500d80b89f467ce4d1514f37c14ec/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session33.pdf" target="[object Object]">Chapter 19.1–19.3 (PDF)</a> of <em>Mathematics for Computer Science</em> for 4.6 Deviation: Markov & Chebyshev Bounds.</p>
<p>View the <a href="/assets/courseware/v1/b9170e02e870aa02f242a70b66f0cf34/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp33.pdf" target="[object Object]">Section 4.6 In-Class Questions (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Deviation from the Mean</h2>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/6c438e3ea6241de145cd8c1eb4852104/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_DeviatTheMean.pdf" target="[object Object]">Deviation from the Mean (PDF)<br /></a><a href="/assets/courseware/v1/b7d14d02f8f56fc0fd751c6d3fefd7ce/asset-v1:OCW+6.042J+2T2019+type@asset+block/DeviationFromtheMean_4.6_Lectrans.pdf" target="[object Object]"></a></p>
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<h2 class="hd hd-2 unit-title">Exercise | Don't Expect the Expectation</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Markov Bounds Practice</h2>
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Markov Bound
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<label class="problem-group-label" for="input_0bf9648eedaa4f128153a2c13ef8a676_2_1" id="label_0bf9648eedaa4f128153a2c13ef8a676_2_1">Given a random variable <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> with <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>R</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mn>50</mn>
</math>, give the smallest value <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>x</mi>
</math> such that the Markov Bound guarantees <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>R</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mi>x</mi>
<mo stretchy="false">]</mo>
<mo>&#8804;<!-- &#8804; --></mo>
<mn>0.5</mn>
</math> under the following conditions.
<br/>
<br/>
<em> If the Markov bound cannot be applied, enter 6042.</em>
<br/>
<br/>
1. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> is non-negative</label>
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<label class="problem-group-label" for="input_0bf9648eedaa4f128153a2c13ef8a676_3_1" id="label_0bf9648eedaa4f128153a2c13ef8a676_3_1">2. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>50</mn>
</math></label>
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<label class="problem-group-label" for="input_0bf9648eedaa4f128153a2c13ef8a676_4_1" id="label_0bf9648eedaa4f128153a2c13ef8a676_4_1">3. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>30</mn>
</math></label>
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<label class="problem-group-label" for="input_0bf9648eedaa4f128153a2c13ef8a676_5_1" id="label_0bf9648eedaa4f128153a2c13ef8a676_5_1">4. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>P</mi>
<mi>r</mi>
<mo stretchy="false">[</mo>
<mi>R</mi>
<mo>&#8804;<!-- &#8804; --></mo>
<mn>50</mn>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mn>0.7</mn>
</math></label>
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<h2 class="hd hd-2 unit-title">Lecture Video | Chebyshev Bounds</h2>
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Recent studies have shown that some small area in our brains is sensitive to <em>nonsensical structures</em>: the 1,000,000 neurons of that area seem to be firing whenever the individual is presented with such structures. This area is now known as the Grader's Area.
<br/>
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In a now famous series of experiments (Pseudoneurologica, 28(1):123-129, 1999), several 6.042 TAs have been presented with students' solutions to exam problems. On reading a solution, the <em>expected number</em> of neurons to fire in the Grader's Area was estimated to be 550,000. Assume that this estimate is indeed correct.
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Moreover, a TA realizes that a solution is <em>bad</em> if the number of neurons in his/her Grader's Area that fire is greater than the number of neurons that do not by 200,000.
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<mi>X</mi>
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<mo stretchy="false">(</mo>
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<h2 class="hd hd-2 unit-title">Exercise | Flipping Coins</h2>
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Flipping Coins
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p> Suppose you flip a fair coin 100 times. The coin flips are all mutually independent.
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><p>2. What is an upper bound on the probability that the number of heads is at least 70 according to Markov's Theorem? <em>Please answer as a fraction in the form x/y.</em>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><p>3. What is the variance of the number of heads? </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><p>4. What is an upper bound on the probability that the number of heads is less than 30 or greater than 70 according to Chebyshev's Theorem? <em>Please answer as a fraction in the form x/y.</em></p>
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<h2 class="hd hd-2 unit-title">Exercise | Practice with Bounds</h2>
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Practice with Bounds
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Suppose 120 students take the 6.042 final exam and the mean of their grades is 90. However, you have no other information about the students and the exam, for example, you should not assume that the final is worth 100 points.
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<label class="problem-group-label" for="input_012751c4dfbf421c8724806568ab4c2f_3_1" id="label_012751c4dfbf421c8724806568ab4c2f_3_1">2. Now suppose somebody tells you that the lowest score on the exam is 30. Compute the new best possible upper bound on the <strong>number of students</strong> who scored at least 180.</label>
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<h2 class="hd hd-2 unit-title">Exercise | Implications of Expectation</h2>
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Implications of Expectation
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p> Complete the following statements so that they are guaranteed to be true.
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<legend id="c6facd889da043cd9830fae29ae1ab44_2_1-legend" class="response-fieldset-legend field-group-hd">Suppose <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>X</mi>
</math> is a nonnegative integer random variable for which <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
</math>=5. You know nothing else about <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>X</mi>
</math>.
<br/>
<br/>
1. The probability that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>X</mi>
<mo>=</mo>
<mn>5</mn>
</math> is</legend>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_2_1" id="input_c6facd889da043cd9830fae29ae1ab44_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="c6facd889da043cd9830fae29ae1ab44_2_1-choice_0-label" for="input_c6facd889da043cd9830fae29ae1ab44_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_2_1"> positive
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_2_1" id="input_c6facd889da043cd9830fae29ae1ab44_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c6facd889da043cd9830fae29ae1ab44_2_1-choice_1-label" for="input_c6facd889da043cd9830fae29ae1ab44_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_2_1"> less than 1
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_2_1" id="input_c6facd889da043cd9830fae29ae1ab44_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c6facd889da043cd9830fae29ae1ab44_2_1-choice_2-label" for="input_c6facd889da043cd9830fae29ae1ab44_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_2_1"> at least <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_2_1" id="input_c6facd889da043cd9830fae29ae1ab44_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="c6facd889da043cd9830fae29ae1ab44_2_1-choice_3-label" for="input_c6facd889da043cd9830fae29ae1ab44_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_2_1"> at most <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_2_1" id="input_c6facd889da043cd9830fae29ae1ab44_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="c6facd889da043cd9830fae29ae1ab44_2_1-choice_4-label" for="input_c6facd889da043cd9830fae29ae1ab44_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_2_1"> could be 0
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_2_1" id="input_c6facd889da043cd9830fae29ae1ab44_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="c6facd889da043cd9830fae29ae1ab44_2_1-choice_5-label" for="input_c6facd889da043cd9830fae29ae1ab44_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_2_1"> 1
</label>
</div>
<span id="answer_c6facd889da043cd9830fae29ae1ab44_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_c6facd889da043cd9830fae29ae1ab44_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="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c6facd889da043cd9830fae29ae1ab44_3_1">
<fieldset aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_3_1">
<legend id="c6facd889da043cd9830fae29ae1ab44_3_1-legend" class="response-fieldset-legend field-group-hd">2. With positive probability, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>X</mi>
</math> is</legend>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_3_1" id="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="c6facd889da043cd9830fae29ae1ab44_3_1-choice_0-label" for="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_3_1"> equal to 0
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_3_1" id="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c6facd889da043cd9830fae29ae1ab44_3_1-choice_1-label" for="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_3_1"> at most <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_3_1" id="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c6facd889da043cd9830fae29ae1ab44_3_1-choice_2-label" for="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_3_1"> less than 5
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_3_1" id="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="c6facd889da043cd9830fae29ae1ab44_3_1-choice_3-label" for="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_3_1"> at most 5
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_3_1" id="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="c6facd889da043cd9830fae29ae1ab44_3_1-choice_4-label" for="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_3_1"> equal to 5
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_3_1" id="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="c6facd889da043cd9830fae29ae1ab44_3_1-choice_5-label" for="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_3_1"> great than 5
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_3_1" id="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="c6facd889da043cd9830fae29ae1ab44_3_1-choice_6-label" for="input_c6facd889da043cd9830fae29ae1ab44_3_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_3_1"> at least 10
</label>
</div>
<span id="answer_c6facd889da043cd9830fae29ae1ab44_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_c6facd889da043cd9830fae29ae1ab44_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c6facd889da043cd9830fae29ae1ab44_4_1">
<fieldset aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_4_1">
<legend id="c6facd889da043cd9830fae29ae1ab44_4_1-legend" class="response-fieldset-legend field-group-hd">3. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mo stretchy="false">[</mo>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">]</mo>
</math></legend>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_4_1" id="input_c6facd889da043cd9830fae29ae1ab44_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="c6facd889da043cd9830fae29ae1ab44_4_1-choice_0-label" for="input_c6facd889da043cd9830fae29ae1ab44_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_4_1"> is 25
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_4_1" id="input_c6facd889da043cd9830fae29ae1ab44_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c6facd889da043cd9830fae29ae1ab44_4_1-choice_1-label" for="input_c6facd889da043cd9830fae29ae1ab44_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_4_1"> is not 25
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_4_1" id="input_c6facd889da043cd9830fae29ae1ab44_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c6facd889da043cd9830fae29ae1ab44_4_1-choice_2-label" for="input_c6facd889da043cd9830fae29ae1ab44_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_4_1"> is at most 25
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_4_1" id="input_c6facd889da043cd9830fae29ae1ab44_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="c6facd889da043cd9830fae29ae1ab44_4_1-choice_3-label" for="input_c6facd889da043cd9830fae29ae1ab44_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_4_1"> is at most <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_4_1" id="input_c6facd889da043cd9830fae29ae1ab44_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="c6facd889da043cd9830fae29ae1ab44_4_1-choice_4-label" for="input_c6facd889da043cd9830fae29ae1ab44_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_4_1"> could be 0
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_4_1" id="input_c6facd889da043cd9830fae29ae1ab44_4_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="c6facd889da043cd9830fae29ae1ab44_4_1-choice_5-label" for="input_c6facd889da043cd9830fae29ae1ab44_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_4_1"> could be 100
</label>
</div>
<span id="answer_c6facd889da043cd9830fae29ae1ab44_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_c6facd889da043cd9830fae29ae1ab44_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c6facd889da043cd9830fae29ae1ab44_5_1">
<fieldset aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_5_1">
<legend id="c6facd889da043cd9830fae29ae1ab44_5_1-legend" class="response-fieldset-legend field-group-hd">4. The probability <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>1000</mn>
<mo stretchy="false">]</mo>
</math></legend>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_5_1" id="input_c6facd889da043cd9830fae29ae1ab44_5_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="c6facd889da043cd9830fae29ae1ab44_5_1-choice_0-label" for="input_c6facd889da043cd9830fae29ae1ab44_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_5_1"> zero
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_5_1" id="input_c6facd889da043cd9830fae29ae1ab44_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c6facd889da043cd9830fae29ae1ab44_5_1-choice_1-label" for="input_c6facd889da043cd9830fae29ae1ab44_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_5_1"> positive
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_5_1" id="input_c6facd889da043cd9830fae29ae1ab44_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c6facd889da043cd9830fae29ae1ab44_5_1-choice_2-label" for="input_c6facd889da043cd9830fae29ae1ab44_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_5_1"> at most <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>1</mn>
<mn>1000</mn>
</mfrac>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_5_1" id="input_c6facd889da043cd9830fae29ae1ab44_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="c6facd889da043cd9830fae29ae1ab44_5_1-choice_3-label" for="input_c6facd889da043cd9830fae29ae1ab44_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_5_1"> at most <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>1</mn>
<mn>500</mn>
</mfrac>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_5_1" id="input_c6facd889da043cd9830fae29ae1ab44_5_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="c6facd889da043cd9830fae29ae1ab44_5_1-choice_4-label" for="input_c6facd889da043cd9830fae29ae1ab44_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_5_1"> at most <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>1</mn>
<mn>200</mn>
</mfrac>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_c6facd889da043cd9830fae29ae1ab44_5_1" id="input_c6facd889da043cd9830fae29ae1ab44_5_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="c6facd889da043cd9830fae29ae1ab44_5_1-choice_5-label" for="input_c6facd889da043cd9830fae29ae1ab44_5_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_c6facd889da043cd9830fae29ae1ab44_5_1"> at most <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>5</mn>
<mo>&#8901;<!-- &#8901; --></mo>
<mn>1000</mn>
<mo stretchy="false">]</mo>
</math>
</label>
</div>
<span id="answer_c6facd889da043cd9830fae29ae1ab44_5_1"/>
</fieldset>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<span id="solution_c6facd889da043cd9830fae29ae1ab44_solution_1"/>
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<span id="solution_c6facd889da043cd9830fae29ae1ab44_solution_3"/>
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<span id="solution_c6facd889da043cd9830fae29ae1ab44_solution_4"/>
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