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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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Problem 1
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<p>Suppose a fair die is tossed ten consecutive times. Consider these two possible outcomes:</p>
<p style="padding-left:60px;"><b>Outcome 1:</b> The ten tosses result in the sequence \(\langle 6, 6, 6, 6, 6, 6, 6, 6, 6, 6\rangle\) (in that order).</p>
<p style="padding-left:60px;"><b>Outcome 2:</b> The ten tosses result in the sequence \(\langle 6, 1, 1, 6, 1, 6, 1, 6, 1, 1\rangle\) (in that order).</p>
<p>Which outcome is more probable?</p>
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</div><p><br/>Next, suppose you are dealt a hand of ten cards from a standard deck of \(52\) cards, half of which are red (\(R\)) and half of which are black (\(B\)). Consider these two possible outcomes:</p>
<p style="padding-left:60px;"><b>Outcome X:</b> You are dealt a sequence of the form \(\langle B, B, B, B, B, B, B, B, B, B\rangle\) (in that order).</p>
<p style="padding-left:60px;"><b>Outcome Y:</b> You are dealt a sequence of the form \(\langle B, R, R, B, R, B, R, B, R, R\rangle\) (in that order).</p>
<p>Which outcome is more probable?</p>
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Problem 2
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<p> Suppose you are about to be dealt a two-card hand from a standard deck of cards (which consists of 52 cards, four of which are aces and thirteen of which are diamonds). The cards are picked totally at random.</p>
<p>What credence should you assign to the proposition that your two-card hand will contain at least one ace? (Represent your answer as a fraction of whole numbers.)</p>
<p>(If you have not attempted a problem like this before, you might find online resources such as <a href="http://www.wikihow.com/Calculate-Probability">this one</a> helpful.)</p>
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</div><p><br/>Suppose again that you are dealt a two-card hand from a standard deck of cards (13 out of 52 of which are diamonds). And again, the cards are picked totally at random. This time, you are offered a bet. If your hand contains at least one diamond, you get \($100\). If it doesn&#8217;t, you lose \($10\).</p>
<p>Assuming you value \($n\) to degree \(n\), what is the expected value of taking the bet? (Represent your answer as a fraction of whole numbers.)</p>
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Problem 3
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<p>Recall the following connection between objective and subjective probability:</p>
<p style="padding-left:60px;"><i>The Objective-Subjective Connection</i>: The objective probability of \(A\) at time \(t\) is the subjective probability that a perfectly rational agent would assign to \(A\), if she had perfect information about events before \(t\) and no information about events after \(t\).</p>
<p>And recall that a <i>deterministic world</i> is such that given a full specification of the world at any given time, the laws can be used to determine a full specification of the world at any future time.</p>
<p>Now, suppose that we live in a deterministic world, and that I am about to toss an ordinary coin. Assuming that the Objective-Subjective Connection holds, what is the <i>objective</i> probability that the toss will land heads?</p>
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<input type="radio" name="input_f1ba00ab48964198994ca00ec66cc10a_2_1" id="input_f1ba00ab48964198994ca00ec66cc10a_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="f1ba00ab48964198994ca00ec66cc10a_2_1-choice_0-label" for="input_f1ba00ab48964198994ca00ec66cc10a_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_f1ba00ab48964198994ca00ec66cc10a_2_1"> \(0\)
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<input type="radio" name="input_f1ba00ab48964198994ca00ec66cc10a_2_1" id="input_f1ba00ab48964198994ca00ec66cc10a_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="f1ba00ab48964198994ca00ec66cc10a_2_1-choice_1-label" for="input_f1ba00ab48964198994ca00ec66cc10a_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_f1ba00ab48964198994ca00ec66cc10a_2_1"> \(.5\)
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<input type="radio" name="input_f1ba00ab48964198994ca00ec66cc10a_2_1" id="input_f1ba00ab48964198994ca00ec66cc10a_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="f1ba00ab48964198994ca00ec66cc10a_2_1-choice_2-label" for="input_f1ba00ab48964198994ca00ec66cc10a_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_f1ba00ab48964198994ca00ec66cc10a_2_1"> \(1\)
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<input type="radio" name="input_f1ba00ab48964198994ca00ec66cc10a_2_1" id="input_f1ba00ab48964198994ca00ec66cc10a_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="f1ba00ab48964198994ca00ec66cc10a_2_1-choice_3-label" for="input_f1ba00ab48964198994ca00ec66cc10a_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_f1ba00ab48964198994ca00ec66cc10a_2_1"> Either \(0\) or \(1\)
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</div><p><br/>What <i>subjective</i> probability should you assign to the the proposition that the toss will land heads? (Assume you're in an ordinary situation and don't know the exact details of the forces acting on the coin toss.)</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_f1ba00ab48964198994ca00ec66cc10a_3_1">
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<input type="radio" name="input_f1ba00ab48964198994ca00ec66cc10a_3_1" id="input_f1ba00ab48964198994ca00ec66cc10a_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="f1ba00ab48964198994ca00ec66cc10a_3_1-choice_0-label" for="input_f1ba00ab48964198994ca00ec66cc10a_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_f1ba00ab48964198994ca00ec66cc10a_3_1"> \(0\)
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<input type="radio" name="input_f1ba00ab48964198994ca00ec66cc10a_3_1" id="input_f1ba00ab48964198994ca00ec66cc10a_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="f1ba00ab48964198994ca00ec66cc10a_3_1-choice_1-label" for="input_f1ba00ab48964198994ca00ec66cc10a_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_f1ba00ab48964198994ca00ec66cc10a_3_1"> \(.5\)
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<input type="radio" name="input_f1ba00ab48964198994ca00ec66cc10a_3_1" id="input_f1ba00ab48964198994ca00ec66cc10a_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="f1ba00ab48964198994ca00ec66cc10a_3_1-choice_2-label" for="input_f1ba00ab48964198994ca00ec66cc10a_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_f1ba00ab48964198994ca00ec66cc10a_3_1"> \(1\)
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<input type="radio" name="input_f1ba00ab48964198994ca00ec66cc10a_3_1" id="input_f1ba00ab48964198994ca00ec66cc10a_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="f1ba00ab48964198994ca00ec66cc10a_3_1-choice_3-label" for="input_f1ba00ab48964198994ca00ec66cc10a_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_f1ba00ab48964198994ca00ec66cc10a_3_1"> Either \(0\) or \(1\)
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Problem 4
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<p>You are offered a chance to play a game that works as follows. A coin will be tossed ten times. If all ten tosses are Tails, you get \($100,000\). If not, you get \($2^n\) if the coin first lands Heads on the \(n\)th toss.</p>
<p>Assuming you value \($n\) to degree \(n\), what is the expected value of playing this game? (Represent your answer as a fraction of whole numbers.)</p>
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\(\)
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