Past exam of the year 1 of the computer science course of the University of Oxford / 2003 / Calculus, probability, linear algebra, discrete mathematics, logic and proof, and models of computation / 5 / b / ii

Let $I$ be the number of heads in the initial allocation tosses. Then $I$ has the binomial distribution with parameters $n$ and $1/2$. Given $I=i$, the winning probability is $i/n$, including the already finished games $i=0,n$. The law of total probability and linearity of expectation giveThus, for $n\ge 1$,