= Solution
{c}
For a <geometric distribution> on positive integers, the <probability generating function> is $p s/[1-(1-p)s]$. Conditional averaging over the red die yields
$$
\boxed{G_T(s)=\frac16\sum_{r=1}^6\frac{(7-r)s}{6-(r-1)s}
=\frac16\sum_{k=1}^6\frac{ks}{6-(6-k)s}.}
$$
As a <power series>, its exact domain of absolute convergence is
$$
\boxed{|s|<\frac65.}
$$
Indeed the slowest decaying component has failure <probability> $5/6$, so $\mathbb P(T=t)\ge(1/36)(5/6)^{t-1}$. At $|s|=6/5$, the terms of the <power series> fail to tend to zero; outside that disk they grow. All six component series converge inside it. In particular, the <probability generating function> is defined on the usual real <interval> $0\le s\le1$, with $G_T(1)=1$. The rational expression beyond the convergence disk is only an algebraic continuation, not the defining <expected value>.
Back to article page
