Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability resit / 7 / b / ii / Solution

Write $G(s)=\mathbb{E}[s^F]$. The initial possibilities are $H$ with probability $1/2$, after which a fresh wait begins; $TH$ with probability $1/4$, after which a fresh wait begins; or $TT$ with probability $1/4$, which ends the experiment. Accounting for the flips already made in each case gives the probability generating function equationTherefore $G_F(s)=\frac{s^2}{4-2s-s^2}$ for $0\le s\le 1$. In particular, $G_F(1)=1$ and there is no mass below two flips, as required.