Chapter 2.2, Problem 9E

In the gambling game chuck-a-luck, for a $1 bet it is possible to win $1, $2, or $3 with respective probabilities $\frac{75}{216},\frac{15}{216}$, and $\frac{1}{216}$.

One dollar is lost with probability $\frac{125}{216}$.

Let X equal the payoff for this game and find E(X). Note that when a bet is won, the $1 that was bet, in addition to the $1, $2, or $3 that is won, is returned to the bettor.