Chapter 2, Problem 2.25P

A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability that a 5 occurs first. Hint: Let $E_n$ denote the event that a 5 occurs on the nth roll and no 5 or 7 occurs on the first $n-1$ rolls. Compute $P\left(E_n\right)$ and argue that ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}P\left(E_n\right)}$ is the desired probability.