Expected value

====Examples====
[[File:Largenumbers.svg|thumb|An illustration of the convergence of sequence averages of rolls of a dice to the expected value of 3.5 as the number of rolls (trials) grows]]

* Let <math>X</math> represent the outcome of a roll of a fair six-sided {{dice}}. More specifically, <math>X</math> will be the number of [[Pip (counting)|pips]] showing on the top face of the {{dice}} after the toss. The possible values for <math>X</math> are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of {{frac2|1|6}}. The expectation of <math>X</math> is
::<math>\operatorname{E}[X] = 1\cdot\frac16 + 2\cdot\frac16 + 3\cdot\frac16 + 4\cdot\frac16 + 5\cdot\frac16 + 6\cdot\frac16 = 3.5.</math>
:If one rolls the {{dice}} <math>n</math> times and computes the average ([[arithmetic mean]]) of the results, then as <math>n</math> grows, the average will [[almost surely]] [[Convergent sequence|converge]] to the expected value, a fact known as the [[strong law of large numbers]].

* The [[roulette]] game consists of a small ball and a wheel with 38 numbered pockets around the edge. As the wheel is spun, the ball bounces around randomly until it settles down in one of the pockets. Suppose random variable <math>X</math> represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability {{frac2|1|38}} in American roulette), the payoff is $35; otherwise the player loses the bet. The expected profit from such a bet will be
::<math>\operatorname{E}[\,\text{gain from }\$1\text{ bet}\,] = -\$1 \cdot \frac{37}{38} + \$35 \cdot \frac{1}{38} = -\$\frac{1}{19}.</math>
:That is, the expected value to be won from a $1 bet is −${{frac2|1|19}}. Thus, in 190 bets, the net loss will probably be about $10.