102 lines
2.5 KiB
TeX
Executable File
102 lines
2.5 KiB
TeX
Executable File
\section{Dice}
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\definition{}
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A \textit{die} is a device that randomly selects a positive integer from
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a finite list of options. For example, the standard 6-sided die selects a value from
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$[1,2,3,4,5,6]$. We may have many sides with the same value, as in $[1, 1, 2, 3]$.
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To describe a die with a generating function, let $a_k$ be the number of times
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$k$ appears as a side of the die and consider $a_0 + a_1x + x_2x^2 + ... $. \par
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A die has a finite number of sides, so this will be a regular polynomial.
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\problem{}
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What is the generating function of the standard 6-sided die?
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\begin{solution}
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$x + x^2 + x^3 + x^4 + x^5 + x^6$
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\end{solution}
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\vfill
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\problem{}
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What is the generating function of the die with sides $[1, 2, 3, 5]$?
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\begin{solution}
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$2x + x^2 + x^3 + x^5$
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\end{solution}
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\vfill
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\problem{}
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Let $A(x)$ and $B(x)$ be the generating functions of two dice. \par
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What is the significance of $A(1)$?
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\begin{solution}
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$A(1) = $ the number of sides on the die
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\end{solution}
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\vfill
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\problem{}
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Using formulas we found earlier, show that the $k^\text{th}$ coefficient
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of $A(x)B(x)$ is the number of ways to roll $k$ as the sum of the two dice.
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\begin{solution}
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The $k^\text{th}$ coefficient of $A(x)B(x)$ is...
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\begin{align*}
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a_0b_k + a_1b_{k+1} + ... + a_kb_0 \\
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&=~ \text{count}(A = 0; B = k) + ... + \text{count}(A = k; B = 0) \\
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&=~ \text{number of ways} A + B = k
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\end{align*}
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\end{solution}
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\vfill
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\pagebreak
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\problem{}
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Find a generating function for the sequence $c_0, c_1, ...$, where $c_k$ is
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the probability that the sum of the two dice is $k$.
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\begin{solution}
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\begin{equation*}
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c_k
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= \frac{\text{number of ways sum } = k}{\text{number of total outcomes}}
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= \frac{\text{number of ways sum } = k}{A(1)B(1)}
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\end{equation*}
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So,
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\begin{equation*}
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c_0 + c_1x + c_2x^2 =
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\frac{A(x)B(x)}{A(1)B(1)}
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\end{equation*}
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\end{solution}
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\vfill
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\problem{}
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Using generating functions, find two six-sided dice whose sum has the same
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distribution as the sum of two standard six-sided dice? \par
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That is, for any integer $k$, the number if ways that the sum of the two
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nonstandard dice rolls as $k$ is equal to the numer of ways the sum of
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two standard dice rolls as $k$.
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\hint{factor polynomials.}
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\begin{solution}
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We need a different factorization of
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\begin{equation*}
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(x + x^2 + x^3 + x^4 + x^5 + x^6)^2 = A(x)B(x)
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\end{equation*}
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We can use
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\begin{equation*}
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(x + 2x^2 + 2x^3 + x^4)
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(x + x^3 + x^4 + x^5 + x^6 + x^8)
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\end{equation*}
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\end{solution}
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\vfill
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\pagebreak |