Two players A and B play a game where they take turns rolling a fair die. The first player to roll a $6$ wins. If Agoes first, what is the probability that A wins?

Solution

1. Let $p=1/6$ be the probability of rolling a $6$.
2. A wins if they roll a $6$ on turn $1,3,5,\dots$.
3. The probability is $p+(1-p{)}^{2}p+(1-p{)}^{4}p+\dots$.
4. This is a geometric series with first term $1/6$ and common ratio $(5/6{)}^2=25/36$.
5. The sum is $$\begin{array}{rl}(1/6)/(1-25/36)& =(1/6)/(11/36)\\ & =6/11\end{array}$$.
6. Hence the answer is (B).