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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/8fe137a1e569a7c8a04af6077836c465/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session34.pdf" target="[object Object]">Chapter 19.4–19.5 (PDF)</a> of <em>Mathematics for Computer Science</em> for 4.7 Sampling & Confidence.</p>
<p>View the <a href="/assets/courseware/v1/b7be7d68c4152657a6f9dbc9e4aebab4/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp34.pdf" target="[object Object]">Section 4.7 In-Class Questions (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Law of Large Numbers</h2>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/e607ed8296e524ea79d4ceaa8e0b67cd/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_LawLrgeNumbr.pdf" target="[object Object]">Law of Large Numbers (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Exercise | Not So Strong</h2>
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Not So Strong
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<label class="problem-group-label" for="input_e86d0403afa6499cb402629551dbc79e_2_1" id="label_e86d0403afa6499cb402629551dbc79e_2_1">For each positive integer <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>i</mi>
</math> let the random variable <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
</math> be defined as:
<br/>
<br/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mlabeledtr>
<mtd id="mjx-eqn-1">
<mtext>(1)</mtext>
</mtd>
<mtd>
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>=</mo>
<mo>&#177;<!-- &#177; --></mo>
<mi>i</mi>
<mo>&#8901;<!-- &#8901; --></mo>
<msup>
<mn>2</mn>
<mi>i</mi>
</msup>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo stretchy="false">)</mo>
</mtd>
</mlabeledtr>
<mlabeledtr>
<mtd id="mjx-eqn-2">
<mtext>(2)</mtext>
</mtd>
<mtd>
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mn>1</mn>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>2</mn>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo stretchy="false">)</mo>
</mtd>
</mlabeledtr>
</mtable>
</math>where <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>0</mn>
<mo>&#8804;<!-- &#8804; --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo stretchy="false">)</mo>
<mo>&#8804;<!-- &#8804; --></mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math>
<br/>
<br/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>S</mi>
<mi>n</mi>
</msub>
<mo>:=</mo>
<msub>
<mi>X</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msub>
<mi>X</mi>
<mn>2</mn>
</msub>
<mo>+</mo>
<mo>&#8230;<!-- &#8230; --></mo>
<mo>+</mo>
<msub>
<mi>X</mi>
<mi>n</mi>
</msub>
</math>
<br/>
<br/>
What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>S</mi>
<mi>n</mi>
</msub>
<mo stretchy="false">]</mo>
</math>?</label>
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<h2 class="hd hd-2 unit-title">Exercise | Sampling Coin Tosses</h2>
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Sampling Coin Tosses
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<label class="problem-group-label" for="input_7b1113d823b049b9852f4477926f83d1_2_1" id="label_7b1113d823b049b9852f4477926f83d1_2_1">We toss <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math> fair coins and count the number of heads.
<br/>
<br/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
</math> be an indicator variable for the <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mi>h</mi>
</mrow>
</msup>
</math> coin. Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>A</mi>
<mi>n</mi>
</msub>
<mo>=</mo>
<mstyle displaystyle="true" scriptlevel="0">
<mfrac>
<mrow>
<munderover>
<mo>&#8721;<!-- &#8721; --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</munderover>
<msub>
<mi>X</mi>
<mi>i</mi>
</msub>
</mrow>
<mi>n</mi>
</mfrac>
</mstyle>
</math>.
<br/>
<br/>
<em>Please format fractional answers as x/y.</em>
<br/>
<br/>
1. What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>A</mi>
<mi>n</mi>
</msub>
<mo stretchy="false">]</mo>
</math>?</label>
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<mi>n</mi>
</math> can be to guarantee <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi>A</mi>
<mi>n</mi>
</msub>
<mo>&#8722;<!-- &#8722; --></mo>
<mi>E</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>A</mi>
<mi>n</mi>
</msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>&gt;</mo>
<mfrac>
<mn>1</mn>
<mn>10</mn>
</mfrac>
<mo stretchy="false">]</mo>
<mo>&#8804;<!-- &#8804; --></mo>
<mfrac>
<mn>1</mn>
<mn>25</mn>
</mfrac>
</math>?</label>
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<label class="problem-group-label" for="input_7b1113d823b049b9852f4477926f83d1_4_1" id="label_7b1113d823b049b9852f4477926f83d1_4_1">3. What is the smallest <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>&#948;<!-- &#948; --></mi>
</math> if <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>100</mn>
</mrow>
</msub>
<mo>&#8722;<!-- &#8722; --></mo>
<mi>E</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>100</mn>
</mrow>
</msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>&gt;</mo>
<mi>&#948;<!-- &#948; --></mi>
<mo stretchy="false">]</mo>
<mo>&#8804;<!-- &#8804; --></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math>?</label>
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<h2 class="hd hd-2 unit-title">Lecture Video | Birthday Matching</h2>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/ddf4b50bebf909fd5ffb464ee7c38c58/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Birthday.pdf" target="[object Object]">Birthday Matching (PDF)</a></p>
<p><a href="/assets/courseware/v1/3f5d7efee9267aa9658230067d477991/asset-v1:OCW+6.042J+2T2019+type@asset+block/BirthdayMatching_4.7_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Exercise | Birthdays on Naboo</h2>
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Birthdays on Naboo
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<h2 class="hd hd-2 unit-title">Lecture Video | Sampling & Confidence</h2>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/1bfcfac53cb8a9e98810022963009beb/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_SmplingConfid.pdf" target="[object Object]">Sampling and Confidence (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Exercise | Confidence</h2>
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Confidence
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<ns0:mo>&#177;</ns0:mo>
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How many samples would be required for the following scenarios, with all other parameters held constant?</p>
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<h2 class="hd hd-2 unit-title">Exercise | Random Sampling</h2>
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Random Sampling
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<legend id="e3491c1616414f5da9f87233bcb318f5_2_1-legend" class="response-fieldset-legend field-group-hd">You work for the president and you want to estimate the fraction <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>p</mi></math> of voters in the entire nation that will prefer him in the upcoming elections. You do this by random sampling. Specifically, you select <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi></math> voters independently and randomly, ask them who they are going to vote for, and use the fraction <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>P</mi></math> of those that say they will vote for the President as an estimate for <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi></math>.
<br/>
<br/>
1. Our theorems about sampling and distributions allow us to calculate how confident we can be that the random variable <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>P</mi>
</math> takes a value near the constant <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>p</mi>
</math>. This calculation uses some facts about voters and the way they are chosen.
<br/>
<br/>
Which of the following facts are true?</legend>
<div class="field">
<input type="checkbox" name="input_e3491c1616414f5da9f87233bcb318f5_2_1[]" id="input_e3491c1616414f5da9f87233bcb318f5_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="e3491c1616414f5da9f87233bcb318f5_2_1-choice_0-label" for="input_e3491c1616414f5da9f87233bcb318f5_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_e3491c1616414f5da9f87233bcb318f5_2_1"> Given a particular voter, the probability of that voter preferring the President is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>p</mi>
</math>.
</label>
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<input type="checkbox" name="input_e3491c1616414f5da9f87233bcb318f5_2_1[]" id="input_e3491c1616414f5da9f87233bcb318f5_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="e3491c1616414f5da9f87233bcb318f5_2_1-choice_1-label" for="input_e3491c1616414f5da9f87233bcb318f5_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_e3491c1616414f5da9f87233bcb318f5_2_1"> Given a particular voter, the probability of that voter preferring the President is 1 or 0.
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<input type="checkbox" name="input_e3491c1616414f5da9f87233bcb318f5_2_1[]" id="input_e3491c1616414f5da9f87233bcb318f5_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="e3491c1616414f5da9f87233bcb318f5_2_1-choice_2-label" for="input_e3491c1616414f5da9f87233bcb318f5_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_e3491c1616414f5da9f87233bcb318f5_2_1"> The probability that some voter is chosen more than once in the sequence goes to zero as <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math> increases.
</label>
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<input type="checkbox" name="input_e3491c1616414f5da9f87233bcb318f5_2_1[]" id="input_e3491c1616414f5da9f87233bcb318f5_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="e3491c1616414f5da9f87233bcb318f5_2_1-choice_3-label" for="input_e3491c1616414f5da9f87233bcb318f5_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_e3491c1616414f5da9f87233bcb318f5_2_1"> All voters are equally likely to be selected as the third in our sequence of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math> choices of voters (assuming <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>3</mn>
</math>).
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<input type="checkbox" name="input_e3491c1616414f5da9f87233bcb318f5_2_1[]" id="input_e3491c1616414f5da9f87233bcb318f5_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="e3491c1616414f5da9f87233bcb318f5_2_1-choice_4-label" for="input_e3491c1616414f5da9f87233bcb318f5_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_e3491c1616414f5da9f87233bcb318f5_2_1"> The probability that the second voter chosen will favor the President, given that the first voter chosen prefers the President, is greater than <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>p</mi>
</math>.
</label>
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<input type="checkbox" name="input_e3491c1616414f5da9f87233bcb318f5_2_1[]" id="input_e3491c1616414f5da9f87233bcb318f5_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="e3491c1616414f5da9f87233bcb318f5_2_1-choice_5-label" for="input_e3491c1616414f5da9f87233bcb318f5_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_e3491c1616414f5da9f87233bcb318f5_2_1"> The probability that the second voter chosen will favor the President, given that the second voter chosen is from the same state as the first, may not equal <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>p</mi>
</math>.
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</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><p>2. Suppose that, according to your calculations, the following is true about your polling:
</p>
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<legend id="e3491c1616414f5da9f87233bcb318f5_3_1-legend" class="response-fieldset-legend field-group-hd"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mo movablelimits="true" form="prefix">Pr</mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>P</mi>
<mo>&#8722;<!-- &#8722; --></mo>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>&#8804;<!-- &#8804; --></mo>
<mn>0.04</mn>
<mo stretchy="false">]</mo>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>0.95</mn>
</math>
<br/>
You run the poll, you count how many respondents said they will vote for the President, you divide by <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math>, and find 0.53. You call the President, and... what do you say?</legend>
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<input type="checkbox" name="input_e3491c1616414f5da9f87233bcb318f5_3_1[]" id="input_e3491c1616414f5da9f87233bcb318f5_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="e3491c1616414f5da9f87233bcb318f5_3_1-choice_0-label" for="input_e3491c1616414f5da9f87233bcb318f5_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_e3491c1616414f5da9f87233bcb318f5_3_1"> Mr. President, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>p</mi>
</math>=0.53.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_e3491c1616414f5da9f87233bcb318f5_3_1[]" id="input_e3491c1616414f5da9f87233bcb318f5_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="e3491c1616414f5da9f87233bcb318f5_3_1-choice_1-label" for="input_e3491c1616414f5da9f87233bcb318f5_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_e3491c1616414f5da9f87233bcb318f5_3_1"> Mr. President, with probability at least 95%, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>p</mi>
</math> is within 0.04 of 0.53.
</label>
</div>
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<input type="checkbox" name="input_e3491c1616414f5da9f87233bcb318f5_3_1[]" id="input_e3491c1616414f5da9f87233bcb318f5_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="e3491c1616414f5da9f87233bcb318f5_3_1-choice_2-label" for="input_e3491c1616414f5da9f87233bcb318f5_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_e3491c1616414f5da9f87233bcb318f5_3_1"> Mr. President, either <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>p</mi>
</math> is within 0.04 of 0.53 or something very strange (5-in-100) has happened.
</label>
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<input type="checkbox" name="input_e3491c1616414f5da9f87233bcb318f5_3_1[]" id="input_e3491c1616414f5da9f87233bcb318f5_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="e3491c1616414f5da9f87233bcb318f5_3_1-choice_3-label" for="input_e3491c1616414f5da9f87233bcb318f5_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_e3491c1616414f5da9f87233bcb318f5_3_1"> Mr. President, we can be 95% confident that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>p</mi>
</math> is within 0.04 of 0.53.
</label>
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<h2 class="hd hd-2 unit-title">Above Average Number of Fingers</h2>
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<h3 class="hd hd-3 problem-header" id="d75604a0fbf74c67a274ecac6cdb7205-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@d75604a0fbf74c67a274ecac6cdb7205-problem-progress" tabindex="-1">
Above Average Number of Fingers
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>There's a common wisecrack that 90% of drivers consider themselves above average. (A recent mention of this appears in NY Times writer <a href="https://pogue.blogs.nytimes.com/2010/04/29/text-blocking-apps-only-work-if-you-use-them/" target="[object Object]">David Pogue's blog, April 29, 2010.</a>)
This may sound absurd, but maybe not: After all, the vast majority of people have an above average number of fingers.
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More than one of the following assertions explain this fact - which ones?
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<input type="checkbox" name="input_d75604a0fbf74c67a274ecac6cdb7205_2_1[]" id="input_d75604a0fbf74c67a274ecac6cdb7205_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="d75604a0fbf74c67a274ecac6cdb7205_2_1-choice_0-label" for="input_d75604a0fbf74c67a274ecac6cdb7205_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_d75604a0fbf74c67a274ecac6cdb7205_2_1"> Most people have a super secret extra bonus finger of which they are unaware.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_d75604a0fbf74c67a274ecac6cdb7205_2_1[]" id="input_d75604a0fbf74c67a274ecac6cdb7205_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="d75604a0fbf74c67a274ecac6cdb7205_2_1-choice_1-label" for="input_d75604a0fbf74c67a274ecac6cdb7205_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_d75604a0fbf74c67a274ecac6cdb7205_2_1"> A pedantic minority don't count their thumbs as fingers, while the majority of people do.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_d75604a0fbf74c67a274ecac6cdb7205_2_1[]" id="input_d75604a0fbf74c67a274ecac6cdb7205_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="d75604a0fbf74c67a274ecac6cdb7205_2_1-choice_2-label" for="input_d75604a0fbf74c67a274ecac6cdb7205_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_d75604a0fbf74c67a274ecac6cdb7205_2_1"> When you add up the total number of fingers among the world's population and then divide by the size of the population, you get a number less than ten.
</label>
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<div class="field">
<input type="checkbox" name="input_d75604a0fbf74c67a274ecac6cdb7205_2_1[]" id="input_d75604a0fbf74c67a274ecac6cdb7205_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="d75604a0fbf74c67a274ecac6cdb7205_2_1-choice_3-label" for="input_d75604a0fbf74c67a274ecac6cdb7205_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_d75604a0fbf74c67a274ecac6cdb7205_2_1"> This follows from Markov's Theorem, since no one has a negative number of fingers.
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<div class="field">
<input type="checkbox" name="input_d75604a0fbf74c67a274ecac6cdb7205_2_1[]" id="input_d75604a0fbf74c67a274ecac6cdb7205_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="d75604a0fbf74c67a274ecac6cdb7205_2_1-choice_4-label" for="input_d75604a0fbf74c67a274ecac6cdb7205_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_d75604a0fbf74c67a274ecac6cdb7205_2_1"> Missing fingers are much more common than extra ones.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_d75604a0fbf74c67a274ecac6cdb7205_2_1[]" id="input_d75604a0fbf74c67a274ecac6cdb7205_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="d75604a0fbf74c67a274ecac6cdb7205_2_1-choice_5-label" for="input_d75604a0fbf74c67a274ecac6cdb7205_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_d75604a0fbf74c67a274ecac6cdb7205_2_1"> Missing fingers are at least slightly more common than extra ones.
</label>
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