Chapter 2, Problem 2.26P

The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a 2, 3, or 12, the player loses: if the sum is either a 7 or an 11, the player wins. If the outcome is anything else, the player continues to roll the dice until she rolls either the initial outcome or a 7. If the 7 comes first, the player loses, whereas if the initial outcome reoccurs before the 7 appears, the player wins. Compute the probability of a player winning at craps.

Hint: Let E denote the event that the initial outcome is i and the player wins. The desired probability is ${\displaystyle \underset{i=2}{\overset{12}{\sum }}P\left(E_i\right)}$. To compute $P\left(E_i\right)$, define the events $E_{i,n}$, to be the event that the initial sum is i and the player wins on the nth roll. Argue that $P\left(E_i\right)={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}P\left(E_{i,n}\right)}$.