70 lines
1.6 KiB
TeX
70 lines
1.6 KiB
TeX
\section{Introduction}
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\generic{Setup:}
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Suppose we toss a 6-sided die $n$ times. \par
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It is easy to detect the first time we roll a 6. \par
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What should we do if we want to detect the \textit{last}?
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\problem{}<lastl>
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Given $l \leq n$, what is the probability that the last $l$
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tosses of this die contain exactly one six? \par
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\hint{Start with small $l$.}
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\begin{solution}
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$\mathcal{P}(\text{last } l \text{ tosses have exactly one 6}) = (\nicefrac{1}{6})(\nicefrac{5}{6})^{l-1} \times l$
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\end{solution}
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\vfill
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\problem{}
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For what value of $l$ is the probability in \ref{lastl} maximal? \par
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The following table may help. \par
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\note{We only care about integer values of $l$.}
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\begin{center}
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\begin{tabular}{|| c | c | c ||}
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\hline
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\rule{0pt}{3.5mm} % Bonus height for exponent
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$l$ & $(\nicefrac{5}{6})^l$ & $(\nicefrac{1}{6})(\nicefrac{5}{6})^{l}$ \\
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\hline\hline
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0 & 1.00 & 0.167 \\
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\hline
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1 & 0.83 & 0.139 \\
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\hline
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2 & 0.69 & 0.116 \\
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\hline
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3 & 0.58 & 0.096 \\
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\hline
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4 & 0.48 & 0.080 \\
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\hline
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5 & 0.40 & 0.067 \\
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\hline
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6 & 0.33 & 0.056 \\
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\hline
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7 & 0.28 & 0.047 \\
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\hline
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8 & 0.23 & 0.039 \\
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\hline
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\end{tabular}
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\end{center}
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\begin{solution}
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$(\nicefrac{1}{6})(\nicefrac{5}{6})^{l-1} \times l$ is maximal at $l = 5.48$, so $l = 5$. \par
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$l = 6$ is close enough.
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\end{solution}
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\vfill
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\problem{}
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Finish your solution: \par
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In $n$ rolls of a six-sided die, what strategy maximizes
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our chance of detecting the last $6$ that is rolled? \par
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What is the probability of our guess being right?
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\begin{solution}
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Whether $l = 5$, $5.4$, or $6$, the probability of success rounds to $0.40$.
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\end{solution}
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\vfill
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\pagebreak |