Section 7.5

Sums of Dice and Generating Functions

Distributions of sums of dice or other discrete variables by convolution or generating functions: relabelled dice, loaded dice with uniform sums, one set of dice beating another.

Toolkit

Encode a die’s distribution by its probability generating function $G(x)=\sum_k p_k x^k$ ; the pgf of a sum of independent dice is the product of their pgfs, so matching a target sum distribution means factoring (or equating) polynomials. Key facts:

Convolution: $\mathbb{P}(X+Y=n)=\sum_k \mathbb{P}(X=k)\mathbb{P}(Y=n-k)$ , i.e. the coefficient of $x^n$ in $G_X(x)G_Y(x)$ .
A uniform sum over $\{2,\dots,12\}$ forces $G(x)^2 = \frac{1}{11}(x^2+x^3+\cdots+x^{12})$ , which cannot be a product of two valid degree-6 probability polynomials with positive coefficients (AM-GM / endpoint-coefficient obstructions).
Ties and comparisons: for two i.i.d. sums, $\mathbb{P}(X>Y)=\tfrac{1-\mathbb{P}(X=Y)}{2}$ by symmetry, with $\mathbb{P}(X=Y)=\sum_n p_n^2$ .
Self-convolution matching ( $p=p*p$ rescaled) yields Catalan-number coefficients $p_k\propto C_{k-1}p_1^k$ , fixed by normalisation $G(1)=1$ .

Problems