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<h2 class="hd hd-2 unit-title">Problem set 1</h2>
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<p>You can download this pdf file from <a href="/assets/courseware/v1/22d77949e7f310e807186c683ac5b707/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_ps1.pdf">here</a></p><iframe src="/assets/courseware/v1/22d77949e7f310e807186c683ac5b707/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_ps1.pdf" width="100%" height="1000px"> </iframe>
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<h2 class="hd hd-2 unit-title">Pset 1 problem checker</h2>
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<h3 class="hd hd-3 problem-header" id="ps1-1a-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps1-1a-check-problem-progress" tabindex="-1">
Problem 1a
</h3>
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<p>
To check your answers put them in the appropriate box and click the &#8216;Check' button. The checker can do arithmetic. So, 3^2, [mathjaxinline]5/3[/mathjaxinline] and [mathjaxinline]3*4[/mathjaxinline] are okay. </p>
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Unless otherwise stated the checker expects numerical answers to 3 significant digits. </p>
<p>
For formulas you can use R notation. Remember to use a * for multiplication, i.e. [mathjaxinline]3*x[/mathjaxinline] <b class="bf">NOT </b> [mathjaxinline]3x[/mathjaxinline]. </p>
<p>
The checker knows many of the standard functions, e.g. use exp(x) and ln(x) for the exponential and natural log functions. </p>
<p>
The formula checker expects a specific letter for the variables, so be sure to use the ones specified. </p>
<p><b class="bf">1</b> (i) Probability of two pair (give to 3 significant digits): </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps1-1a-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<p>
(ii) Probability of three of a kind (give to 3 significant digits): </p>
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<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps1-1a-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<p>
(iii) Which is more probable? </p>
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<text> Two pair</text>
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<text> Three of a kind</text>
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Problem 2a
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<p><b class="bf">2(a)</b> Give the answers to 3 significant digits. If you have the exact answer as a fraction, you can enter that. </p>
<p>
<p style="display:inline">[mathjaxinline]P[/mathjaxinline](white beats orange) = </p>
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<p>
<p style="display:inline">[mathjaxinline]P[/mathjaxinline](orange beats blue) = </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps1-2a-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<div id="display_ps1-2a-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps1-2a-check_3_1_dynamath" name="input_ps1-2a-check_3_1_dynamath"/>
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</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]P[/mathjaxinline](blue beats white) = </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps1-2a-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
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</span>
<p id="answer_ps1-2a-check_4_1" class="answer"/>
<div id="display_ps1-2a-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps1-2a-check_4_1_dynamath" name="input_ps1-2a-check_4_1_dynamath"/>
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Problem 2b
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<p><b class="bf">2(b)</b> Give the probability that white wins (3 significant digits): </p>
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<p><b class="bf">3(a)</b> If [mathjaxinline]\omega[/mathjaxinline] is an element of [mathjaxinline]S[/mathjaxinline], give the probability [mathjaxinline]P(\omega )[/mathjaxinline] as a function of [mathjaxinline]n[/mathjaxinline]. </p>
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Problem 3bc
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<p>
<b class="bf">3(b)</b>
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</p>
<p><b class="bf">3(c)</b> Give the formula for [mathjaxinline]P(A)[/mathjaxinline] in terms of [mathjaxinline]n[/mathjaxinline]. </p>
<p>
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Problem 3de
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<p>
<b class="bf">3(d)</b>
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</p>
<p><b class="bf">3(e)</b> Give the smallest [mathjaxinline]n[/mathjaxinline] for which [mathjaxinline]P(B) &gt; 0.9[/mathjaxinline]. Note, a simulation only approximates a probability, but in this case there is a clear answer for [mathjaxinline]n[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps1-3de-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<p>
<em>Don't forget to answer the second part of this problem where you run your code using just 30 trials and look at the variability.</em>
</p>
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Problem 3f
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<div class="problem">
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<p><b class="bf">3(f)</b> Because the answer involves factorials and powers which get large quickly, there are numerical issues with actually computing [mathjaxinline]P(B)[/mathjaxinline]. Here we'll check your answer for [mathjaxinline]n=30[/mathjaxinline]. </p>
<p>
To actually compute this you want to first write the formula for [mathjaxinline]\ln (P(B^ C))[/mathjaxinline] &#8211; use the rules of log to write this as a sum of logs. Then compute the answer as [mathjaxinline]1 - \exp (\ln (P(B^ C)))[/mathjaxinline]. </p>
<p>
Conveniently, R has the <tt class="tt">lfactorial(n)</tt> function which computes [mathjaxinline]\ln (n!)[/mathjaxinline]. </p>
<p>
Give [mathjaxinline]P(B)[/mathjaxinline] when [mathjaxinline]n=30[/mathjaxinline] (to 3 significant digits). <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps1-3f-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<p><b class="bf">3(g)</b> As the problem said, the simulation gives an ambiguous answer. The pset checker will accept either of the two adjacent integers that seem plausible. </p>
<p>
Give the smallest [mathjaxinline]n[/mathjaxinline] for which [mathjaxinline]P(C) &gt; 0.5[/mathjaxinline]. <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps1-3g-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
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