To find the expected value of a game, find all of the values associated with the game *v _{1}, v_{2}, …, v_{n}* and the corresponding probabilities

*p*. and add the product of each value and corresponding probability.

_{1}, p_{2}, …, p_{n}*E* = *p*_{1} *v*_{1} + *p*_{2} *v*_{2} + … + *p _{n} v_{n}*

**Problem 14** If you pay $5 to play a game in which a pair of dice is rolled, calculate the expected value of the game given the following outcomes: if a total less than five comes up then you win $20, if a total of 7 or 11 comes up then you win $10 and for all other totals you win nothing.

**Solution** To find the expected value of the game, we need to know all of the possible outcomes and corresponding probabilities.

The expected value is the sum of each outcome times the corresponding probability,

E = 15( ^{6}/_{36 }) + 5( ^{8}/_{36 }) – 5( ^{22}/_{36 }) = ^{20}/_{36} ≈ 0.56

This means that each time you play the game, you can expect to win $0.56.