Expectations

An expectation of value E(X) means the expectation or average

It is calculated through different means Probability has everything to do with expectations

Fair game

The Expected value of a fair game is 0.

Expectation Formula

For every P(x) of a random variable X, we multiply outcome X with the P(X), then add each of these together. a*P(a) + b*P(b) + c*P(c) … = Expectation

Example

A game involves rolling a die. A player who rolls an even number recieves coins equal to 2* roll. For odd, you lose 3*roll. For this game to be a fair game, the expected value must be zero. Is it a fair game?

Roll	value, x	P(x)	x*P(x)
1	-3	1/6	-3/6
2	4	1/6	4/6
3	-9	1/6	-9/6
4	8	1/6	8/6
5	-15	1/6	-15/6
6	12	1/6	12/6
$-\frac{3}{6}+\frac{4}{6}+-\frac{9}{6}+\frac{8}{6}+-\frac{15}{6}+\frac{12}{6}=-\frac{1}{2}$ Not a fair game. Its not zero, its -1/2

For an event that says something like: If you roll a one, roll again, then the value of ‘1’ is zero. Its does not affect the x*P(x)