## 13.2 Outline

- Expected value
- expected value (expectation)
- fair game
- mathematical expectation

- Expectation with a cost of playing
- definition
- roulette wheel and board

## 13.2 Essential Ideas

If an event *E* has several possible outcomes with probabilities *p*_{1}*, p*_{2}*, p*_{3}*, …*, and if for each of these outcomes the amount that can be won is *a*_{1}*, a*_{2}*, a*_{3}*, … *, respectively, then the **mathematical expectation **(or *expected value*) of *E* is

EXPECTATION = *a*_{1}*p*_{1}* + a*_{2}*p*_{2}* + a*_{3}*p*_{3}* + …*

If there is a cost of playing a game, the cost of playing must be subtracted from the expectation.

EXPECTATION = AMT. TO WIN(PROB. OF WINNING) – COST OF PLAYING