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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/1a04b87a77e3bdd4f3e2db9ab066e0de/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session32.pdf" target="[object Object]">Chapter 18.4–18.5 (PDF)</a> of <em>Mathematics for Computer Science</em> for 4.5 Expectation.</p>
<p>View the <a href="/assets/courseware/v1/18f57f5ec790eefa874a23ed22763881/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp32.pdf" target="[object Object]">Section 4.5 In-Class Questions (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Expectation</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Uneven Dice</h2>
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Uneven Dice
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<label class="problem-group-label" for="input_1226140740f340aa94500d60df64df2a_2_1" id="label_1226140740f340aa94500d60df64df2a_2_1">Say I have a six-sided die that lands on an even number with probability <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math>. If it lands on an even number, then it lands on each one with equal probability. The same is true for odd numbers. Find the expected value of this die.
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<h2 class="hd hd-2 unit-title">Lecture Video | Expected Number of Heads</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Expected Number of Heads Practice</h2>
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Expected Number of Heads
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<h2 class="hd hd-2 unit-title">Lecture Video | Total Expectation</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Another Dice and Coin Game</h2>
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Another Dice and Coin Game
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<label class="problem-group-label" for="input_ed9a544456a44a96951e2c43c87ef243_2_1" id="label_ed9a544456a44a96951e2c43c87ef243_2_1">Suppose I flip a fair, two-sided coin. If it comes up heads, then I roll a fair six-sided die until I get an odd number and record the value. Otherwise, I roll until I get an even number and record the value.
Using the law of total expectation, find the expected value of the experiment.
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<h2 class="hd hd-2 unit-title">Exercise | Three Machines Failing</h2>
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Three Machines Failing
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<h2 class="hd hd-2 unit-title">Lecture Video | Linearity of Expectation</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Fair and Biased Coins</h2>
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Fair and Based Coins
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<label class="problem-group-label" for="input_33d69bc0111b4102a7fd4064062dee33_2_1" id="label_33d69bc0111b4102a7fd4064062dee33_2_1">Say I flip 200 coins, 100 of which are fair, and 100 of which land on heads with probability <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math>.
<br/>
What is the expected number of heads?</label>
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<h2 class="hd hd-2 unit-title">Exercise | Binomial Board Breaking</h2>
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Binomial Board Breaking
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p> Bruce Lee is practicing his skills for an unreleased movie by breaking 5 boards with his fists.
Assume he breaks a board with probability 0.8 (he is practicing his left fist; that's why the probability is not 1) and that he breaks each board independently. </p>
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<h2 class="hd hd-2 unit-title">Exercise | Great Expectations</h2>
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Great Expectations
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>1. What is the expected sum of the numbers that come up when you roll a fair 6-sided die and a fair 12-sided die?
<br/>
<br/>
[Assume the faces have values 1-6 and 1-12, respectively.]</p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><p>2. Suppose you have two computers: Computer 1 generates a random number in the set <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML">
<ns0:mo fence="false" stretchy="false">{</ns0:mo>
<ns0:mn>1</ns0:mn>
<ns0:mo>,</ns0:mo>
<ns0:mn>2</ns0:mn>
<ns0:mo>,</ns0:mo>
<ns0:mo>&#8230;</ns0:mo>
<ns0:mo>,</ns0:mo>
<ns0:mn>99</ns0:mn>
<ns0:mo fence="false" stretchy="false">}</ns0:mo>
</ns0:math> with all numbers equally likely. Similarly, Computer 2 generates a random number in <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML">
<ns0:mo fence="false" stretchy="false">{</ns0:mo>
<ns0:mn>1</ns0:mn>
<ns0:mo>,</ns0:mo>
<ns0:mn>2</ns0:mn>
<ns0:mo>,</ns0:mo>
<ns0:mo>&#8230;</ns0:mo>
<ns0:mo>,</ns0:mo>
<ns0:mn>999</ns0:mn>
<ns0:mo fence="false" stretchy="false">}</ns0:mo>
</ns0:math> with all numbers equally likely.
<br/>
<br/>
You roll a fair die, and if a 5 comes up, you generate a random number using Computer 1, otherwise you generate a random number using Computer 2. What is the expected value of the number you generate?</p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><p> 3. Assuming that Computers 1 and 2 act independently, what is the expected value of the product of the numbers they generate? </p>
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<h2 class="hd hd-2 unit-title">Exercise | Expectation of a Uniform Distribution</h2>
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<h3 class="hd hd-3 problem-header" id="6fee110c4fed42de849ca213b1ed03d1-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@6fee110c4fed42de849ca213b1ed03d1-problem-progress" tabindex="-1">
Expectation of a Uniform Distribution
</h3>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_6fee110c4fed42de849ca213b1ed03d1_2_1">
<fieldset aria-describedby="status_6fee110c4fed42de849ca213b1ed03d1_2_1">
<legend id="6fee110c4fed42de849ca213b1ed03d1_2_1-legend" class="response-fieldset-legend field-group-hd">Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>X</mi>
</math> be a random variable with uniform distribution over the integers from <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8722;<!-- &#8722; --></mo>
<mi>n</mi>
</math> to <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>Y</mi>
<mo>:=</mo>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
</math>.
<br/>
<br/>
Which of the following are true?</legend>
<div class="field">
<input type="checkbox" name="input_6fee110c4fed42de849ca213b1ed03d1_2_1[]" id="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="6fee110c4fed42de849ca213b1ed03d1_2_1-choice_0-label" for="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_6fee110c4fed42de849ca213b1ed03d1_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mn>0</mn>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_6fee110c4fed42de849ca213b1ed03d1_2_1[]" id="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="6fee110c4fed42de849ca213b1ed03d1_2_1-choice_1-label" for="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_6fee110c4fed42de849ca213b1ed03d1_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>Y</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mn>0</mn>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_6fee110c4fed42de849ca213b1ed03d1_2_1[]" id="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="6fee110c4fed42de849ca213b1ed03d1_2_1-choice_2-label" for="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_6fee110c4fed42de849ca213b1ed03d1_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>Y</mi>
<mo stretchy="false">]</mo>
<mo>&gt;</mo>
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_6fee110c4fed42de849ca213b1ed03d1_2_1[]" id="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="6fee110c4fed42de849ca213b1ed03d1_2_1-choice_3-label" for="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_6fee110c4fed42de849ca213b1ed03d1_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>+</mo>
<mi>Y</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
<mo>+</mo>
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>Y</mi>
<mo stretchy="false">]</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_6fee110c4fed42de849ca213b1ed03d1_2_1[]" id="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="6fee110c4fed42de849ca213b1ed03d1_2_1-choice_4-label" for="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_6fee110c4fed42de849ca213b1ed03d1_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mi>Y</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>Y</mi>
<mo stretchy="false">]</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_6fee110c4fed42de849ca213b1ed03d1_2_1[]" id="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="6fee110c4fed42de849ca213b1ed03d1_2_1-choice_5-label" for="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_6fee110c4fed42de849ca213b1ed03d1_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>X</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>Y</mi>
</math> are independent variables
</label>
</div>
<div class="field">
<input type="checkbox" name="input_6fee110c4fed42de849ca213b1ed03d1_2_1[]" id="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="6fee110c4fed42de849ca213b1ed03d1_2_1-choice_6-label" for="input_6fee110c4fed42de849ca213b1ed03d1_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_6fee110c4fed42de849ca213b1ed03d1_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mo stretchy="false">[</mo>
<msup>
<mi>Y</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>Y</mi>
<mo stretchy="false">]</mo>
</math>
</label>
</div>
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