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<h2 class="hd hd-2 unit-title">Homework</h2>
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<p><span style="color: #313131;">The exercises below will count towards your grade. </span><strong>You have only one chance to answer these questions</strong>. <span style="color: #313131;">Take your time, and think carefully before answering.</span></p>
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<h3 class="hd hd-3 problem-header" id="c3be4d2e8b69425b9f347252ac0f8796-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@c3be4d2e8b69425b9f347252ac0f8796-problem-progress" tabindex="-1">
Problem 1
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<p>You are considering buying a lottery ticket. Each ticket costs \($2\), and has a \(1\%\) chance of winning. If you win, you get \($500\); otherwise you get nothing. (You care only about money, and assign value \(n\) to a situation in which you net \(n\) dollars.)</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c3be4d2e8b69425b9f347252ac0f8796_2_1">
<fieldset aria-describedby="status_c3be4d2e8b69425b9f347252ac0f8796_2_1">
<legend id="c3be4d2e8b69425b9f347252ac0f8796_2_1-legend" class="response-fieldset-legend field-group-hd">Which of the equations below corresponds to the right expected value calculation?</legend>
<div class="field">
<input type="radio" name="input_c3be4d2e8b69425b9f347252ac0f8796_2_1" id="input_c3be4d2e8b69425b9f347252ac0f8796_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="c3be4d2e8b69425b9f347252ac0f8796_2_1-choice_0-label" for="input_c3be4d2e8b69425b9f347252ac0f8796_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_c3be4d2e8b69425b9f347252ac0f8796_2_1"> \(EV(\)Buying Ticket\() \:= (500)(.01) + (0)(.99)\)
</label>
</div>
<div class="field">
<input type="radio" name="input_c3be4d2e8b69425b9f347252ac0f8796_2_1" id="input_c3be4d2e8b69425b9f347252ac0f8796_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c3be4d2e8b69425b9f347252ac0f8796_2_1-choice_1-label" for="input_c3be4d2e8b69425b9f347252ac0f8796_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c3be4d2e8b69425b9f347252ac0f8796_2_1"> \(EV(\)Buying Ticket\() \:= (500)(.01) + (2)(.99)\)
</label>
</div>
<div class="field">
<input type="radio" name="input_c3be4d2e8b69425b9f347252ac0f8796_2_1" id="input_c3be4d2e8b69425b9f347252ac0f8796_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c3be4d2e8b69425b9f347252ac0f8796_2_1-choice_2-label" for="input_c3be4d2e8b69425b9f347252ac0f8796_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c3be4d2e8b69425b9f347252ac0f8796_2_1"> \(EV(\)Buying Ticket\() \:= (500)(.01) + (-2)(.99)\)
</label>
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<input type="radio" name="input_c3be4d2e8b69425b9f347252ac0f8796_2_1" id="input_c3be4d2e8b69425b9f347252ac0f8796_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="c3be4d2e8b69425b9f347252ac0f8796_2_1-choice_3-label" for="input_c3be4d2e8b69425b9f347252ac0f8796_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c3be4d2e8b69425b9f347252ac0f8796_2_1"> \(EV(\)Buying Ticket\() \:= (498)(.01) + (-2)(.99)\)
</label>
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<span id="answer_c3be4d2e8b69425b9f347252ac0f8796_2_1"/>
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<span id="solution_c3be4d2e8b69425b9f347252ac0f8796_solution_1"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><p>Calculate the expected utility of buying a ticket and enter it here:</p>
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<input type="text" name="input_c3be4d2e8b69425b9f347252ac0f8796_3_1" id="input_c3be4d2e8b69425b9f347252ac0f8796_3_1" data-input-id="c3be4d2e8b69425b9f347252ac0f8796_3_1" value="" aria-describedby="status_c3be4d2e8b69425b9f347252ac0f8796_3_1" size="20"/>
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\(\)
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<span id="solution_c3be4d2e8b69425b9f347252ac0f8796_solution_2"/>
</div><p>Suppose your options are buying the ticket or not buying the ticket. According to the Principle of Expected Value Maximization, should you buy the ticket?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c3be4d2e8b69425b9f347252ac0f8796_4_1">
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<input type="radio" name="input_c3be4d2e8b69425b9f347252ac0f8796_4_1" id="input_c3be4d2e8b69425b9f347252ac0f8796_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="c3be4d2e8b69425b9f347252ac0f8796_4_1-choice_0-label" for="input_c3be4d2e8b69425b9f347252ac0f8796_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_c3be4d2e8b69425b9f347252ac0f8796_4_1"> You should buy the ticket.
</label>
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<input type="radio" name="input_c3be4d2e8b69425b9f347252ac0f8796_4_1" id="input_c3be4d2e8b69425b9f347252ac0f8796_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c3be4d2e8b69425b9f347252ac0f8796_4_1-choice_1-label" for="input_c3be4d2e8b69425b9f347252ac0f8796_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c3be4d2e8b69425b9f347252ac0f8796_4_1"> You should <i>not</i> buy the ticket.
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Problem 2
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<p>There are two boxes before you: Open and Closed. You can see that Open contains \($10\), but cannot see the contents of Closed. You are told, however, that either Closed is completely empty or it contains \($100\).</p>
<p>You are given two choices: one-box or two-box. To one-box is to take the contents of Closed, and leave the contents of Open behind. To two-box is to take the contents of both boxes. The boxes have been filled ahead of time, and their contents will not be changed.</p>
<p>There is no predictor. Instead, a coin was flipped. If it landed Heads, Closed was filled with the \($100\); if it landed Tails, Closed was left empty.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_3c0130681986442c82119c1fc3241a90_2_1">
<fieldset aria-describedby="status_3c0130681986442c82119c1fc3241a90_2_1">
<legend id="3c0130681986442c82119c1fc3241a90_2_1-legend" class="response-fieldset-legend field-group-hd">According to the Principle of Expected Value Maximization, what should you do? Should you one-box or two-box?</legend>
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<input type="radio" name="input_3c0130681986442c82119c1fc3241a90_2_1" id="input_3c0130681986442c82119c1fc3241a90_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="3c0130681986442c82119c1fc3241a90_2_1-choice_0-label" for="input_3c0130681986442c82119c1fc3241a90_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_3c0130681986442c82119c1fc3241a90_2_1"> One-box
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<input type="radio" name="input_3c0130681986442c82119c1fc3241a90_2_1" id="input_3c0130681986442c82119c1fc3241a90_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="3c0130681986442c82119c1fc3241a90_2_1-choice_1-label" for="input_3c0130681986442c82119c1fc3241a90_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_3c0130681986442c82119c1fc3241a90_2_1"> Two-box
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</div><p><br/>Now assume there is a predictor. Yesterday evening, Predictor was enlisted to make a prediction about whether you would one-box or two-box. If Predictor predicted that you would one-box, the \($100\) was placed in Closed. Otherwise, Closed was left empty. Predictor is known to be accurate \(60\%\) of the time. The boxes have now been sealed, and their contents will not be changed.</p>
<p>As before, <i>your decision will have no effect on the contents of Closed</i>. If it now contains the \($100\), it will continue to do so regardless of whether you decide to one-box or two-box.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_3c0130681986442c82119c1fc3241a90_3_1">
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<legend id="3c0130681986442c82119c1fc3241a90_3_1-legend" class="response-fieldset-legend field-group-hd">According to the Principle of Expected Value Maximization, should you one-box or should you two-box?</legend>
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<input type="radio" name="input_3c0130681986442c82119c1fc3241a90_3_1" id="input_3c0130681986442c82119c1fc3241a90_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="3c0130681986442c82119c1fc3241a90_3_1-choice_0-label" for="input_3c0130681986442c82119c1fc3241a90_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_3c0130681986442c82119c1fc3241a90_3_1"> One-box
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<input type="radio" name="input_3c0130681986442c82119c1fc3241a90_3_1" id="input_3c0130681986442c82119c1fc3241a90_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="3c0130681986442c82119c1fc3241a90_3_1-choice_1-label" for="input_3c0130681986442c82119c1fc3241a90_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_3c0130681986442c82119c1fc3241a90_3_1"> Two-box
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Problem 3
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<p>Of each of the conditionals below, say whether it is indicative or not. (Hint: you might find it helpful to review the material in <a href="/courses/course-v1:MITx+24.118x+2T2020/jump_to_id/517cc8cb955d4989b91fdc470fdfb3fe">this section</a>.)</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_7da9483306e04e8dbacbd599fa8ddfe4_2_1">
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<legend id="7da9483306e04e8dbacbd599fa8ddfe4_2_1-legend" class="response-fieldset-legend field-group-hd">If I left the door open, then the cat probably escaped.</legend>
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<legend id="7da9483306e04e8dbacbd599fa8ddfe4_3_1-legend" class="response-fieldset-legend field-group-hd">If Abraham Lincoln had not gone to the theater, then he would not have been assassinated.</legend>
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<input type="radio" name="input_7da9483306e04e8dbacbd599fa8ddfe4_3_1" id="input_7da9483306e04e8dbacbd599fa8ddfe4_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7da9483306e04e8dbacbd599fa8ddfe4_3_1-choice_0-label" for="input_7da9483306e04e8dbacbd599fa8ddfe4_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7da9483306e04e8dbacbd599fa8ddfe4_3_1"> Indicative
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<input type="radio" name="input_7da9483306e04e8dbacbd599fa8ddfe4_3_1" id="input_7da9483306e04e8dbacbd599fa8ddfe4_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7da9483306e04e8dbacbd599fa8ddfe4_3_1-choice_1-label" for="input_7da9483306e04e8dbacbd599fa8ddfe4_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7da9483306e04e8dbacbd599fa8ddfe4_3_1"> Not Indicative
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<legend id="7da9483306e04e8dbacbd599fa8ddfe4_4_1-legend" class="response-fieldset-legend field-group-hd">If you come pick me up from the train station, then I will buy you dinner.</legend>
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<input type="radio" name="input_7da9483306e04e8dbacbd599fa8ddfe4_4_1" id="input_7da9483306e04e8dbacbd599fa8ddfe4_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7da9483306e04e8dbacbd599fa8ddfe4_4_1-choice_0-label" for="input_7da9483306e04e8dbacbd599fa8ddfe4_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7da9483306e04e8dbacbd599fa8ddfe4_4_1"> Indicative
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<input type="radio" name="input_7da9483306e04e8dbacbd599fa8ddfe4_4_1" id="input_7da9483306e04e8dbacbd599fa8ddfe4_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7da9483306e04e8dbacbd599fa8ddfe4_4_1-choice_1-label" for="input_7da9483306e04e8dbacbd599fa8ddfe4_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7da9483306e04e8dbacbd599fa8ddfe4_4_1"> Not Indicative
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<legend id="7da9483306e04e8dbacbd599fa8ddfe4_5_1-legend" class="response-fieldset-legend field-group-hd">If she promised she would be here, then she will be here.</legend>
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<input type="radio" name="input_7da9483306e04e8dbacbd599fa8ddfe4_5_1" id="input_7da9483306e04e8dbacbd599fa8ddfe4_5_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7da9483306e04e8dbacbd599fa8ddfe4_5_1-choice_0-label" for="input_7da9483306e04e8dbacbd599fa8ddfe4_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7da9483306e04e8dbacbd599fa8ddfe4_5_1"> Indicative
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<input type="radio" name="input_7da9483306e04e8dbacbd599fa8ddfe4_5_1" id="input_7da9483306e04e8dbacbd599fa8ddfe4_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7da9483306e04e8dbacbd599fa8ddfe4_5_1-choice_1-label" for="input_7da9483306e04e8dbacbd599fa8ddfe4_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7da9483306e04e8dbacbd599fa8ddfe4_5_1"> Not Indicative
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="choicegroup capa_inputtype" id="inputtype_7da9483306e04e8dbacbd599fa8ddfe4_6_1">
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<legend id="7da9483306e04e8dbacbd599fa8ddfe4_6_1-legend" class="response-fieldset-legend field-group-hd">If you had listened more carefully, then we would not be having this conversation.</legend>
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<input type="radio" name="input_7da9483306e04e8dbacbd599fa8ddfe4_6_1" id="input_7da9483306e04e8dbacbd599fa8ddfe4_6_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7da9483306e04e8dbacbd599fa8ddfe4_6_1-choice_0-label" for="input_7da9483306e04e8dbacbd599fa8ddfe4_6_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7da9483306e04e8dbacbd599fa8ddfe4_6_1"> Indicative
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<input type="radio" name="input_7da9483306e04e8dbacbd599fa8ddfe4_6_1" id="input_7da9483306e04e8dbacbd599fa8ddfe4_6_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7da9483306e04e8dbacbd599fa8ddfe4_6_1-choice_1-label" for="input_7da9483306e04e8dbacbd599fa8ddfe4_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7da9483306e04e8dbacbd599fa8ddfe4_6_1"> Not Indicative
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<h3 class="hd hd-3 problem-header" id="12062139a22c4489b2b2404506511e45-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@12062139a22c4489b2b2404506511e45-problem-progress" tabindex="-1">
Problem 4
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>The following example is due to <a href="https://www.edx.org/bio/caspar-hare" target="_blank">Caspar Hare</a>:</p>
<p style="padding-left:60px;"><b>Symmetrical Worlds</b>:Imagine that the universe is divided in two along a plane. As far as anyone can tell, the universe is completely symmetrical across this plane. Everything that happens on one side appears to be an exact mirror of everything that happens on the other side; there are so far no observed divergences from this pattern. The plane is a great tourist attraction. People go up to the plane to peer into the other side, and invariably see their symmetrical twin peering back at them.</p>
<p style="padding-left:60px;">The plane alternates between being opaque and transparent, and scientists have established that when the plane is opaque, there are no causal interactions between events on different sides of the plane. A favorite activity of visitors is to wait until the plane turns opaque, and unveil some crazy prop or hold some ridiculous pose, only to find their twin displaying the same crazy prop or holding the same ridiculous pose when the plane turns transparent.</p>
<p style="padding-left:60px;">You are having fun with your symmetry twin when then plane turns opaque. A mysterious but credible stranger comes up to you and offers you a deal. "Here is a thousand dollars," she says. "That's yours. But consider this: my symmetrical twin is right now giving your twin, on the other side of the opaque plane, a thousand dollars. If your twin burns her thousand dollars, I will give you a million dollars. Here is a lighter!"</p>
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<legend id="12062139a22c4489b2b2404506511e45_2_1-legend" class="response-fieldset-legend field-group-hd">According to an evidential decision theorist, what is the best way of maximizing your own profits in the Symmetrical World case?</legend>
<div class="field">
<input type="radio" name="input_12062139a22c4489b2b2404506511e45_2_1" id="input_12062139a22c4489b2b2404506511e45_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="12062139a22c4489b2b2404506511e45_2_1-choice_0-label" for="input_12062139a22c4489b2b2404506511e45_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_12062139a22c4489b2b2404506511e45_2_1"> Burn the thousand!
</label>
</div>
<div class="field">
<input type="radio" name="input_12062139a22c4489b2b2404506511e45_2_1" id="input_12062139a22c4489b2b2404506511e45_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="12062139a22c4489b2b2404506511e45_2_1-choice_1-label" for="input_12062139a22c4489b2b2404506511e45_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_12062139a22c4489b2b2404506511e45_2_1"> Don't burn the thousand!
</label>
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<span id="answer_12062139a22c4489b2b2404506511e45_2_1"/>
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<span class="status unanswered" id="status_12062139a22c4489b2b2404506511e45_2_1" data-tooltip="Not yet answered.">
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<span id="solution_12062139a22c4489b2b2404506511e45_solution_1"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_12062139a22c4489b2b2404506511e45_3_1">
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<legend id="12062139a22c4489b2b2404506511e45_3_1-legend" class="response-fieldset-legend field-group-hd">According to an causal decision theorist, what is the best way of maximizing your own profits in the Symmetrical World case?</legend>
<div class="field">
<input type="radio" name="input_12062139a22c4489b2b2404506511e45_3_1" id="input_12062139a22c4489b2b2404506511e45_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="12062139a22c4489b2b2404506511e45_3_1-choice_0-label" for="input_12062139a22c4489b2b2404506511e45_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_12062139a22c4489b2b2404506511e45_3_1"> Burn the thousand!
</label>
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<input type="radio" name="input_12062139a22c4489b2b2404506511e45_3_1" id="input_12062139a22c4489b2b2404506511e45_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="12062139a22c4489b2b2404506511e45_3_1-choice_1-label" for="input_12062139a22c4489b2b2404506511e45_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_12062139a22c4489b2b2404506511e45_3_1"> Don't burn the thousand!
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<span id="solution_12062139a22c4489b2b2404506511e45_solution_2"/>
</div><p>(Hint: you might find it helpful to review the material in <a href="/courses/course-v1:MITx+24.118x+2T2020/jump_to_id/119e939c80c34038a65d3de1a40cb6ca">this section</a>.)</p>
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<span class="notification-message" aria-describedby="12062139a22c4489b2b2404506511e45-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="12062139a22c4489b2b2404506511e45-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="12062139a22c4489b2b2404506511e45-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
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