lean_workbook_plus_13469

Let $f_0$ be the probability that F wins, starting from the beginning of the game. Let $f_T$ be the probability that F wins, given that tails has just been thrown, and let $f_H$ and $f_{HH}$ be the probability that F wins, given that one or two heads have just been thrown, respectively. Then \n\begin{align*}f_0&=\tfrac12(f_T+f_H)\f_T&=\tfrac12(1+f_H)\f_H&=\tfrac12(f_T+f_{HH})\f_{HH}&=\tfrac12(f_T+0)\\end{align*} Thus by substitution $f_T=\tfrac12+\tfrac12(\tfrac12f_T+\tfrac14f_T)\implies \tfrac58f_T=\tfrac12$ from which we get $f_T=\tfrac45,\ f_H=\tfrac35\implies f_0=\tfrac7{10}$ .