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Co-authored-by: Mark <mark@betalupi.com>
Co-committed-by: Mark <mark@betalupi.com>

src/Advanced/Generating Functions/parts/02 dice.tex

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