# Expected value

In probability, the expected value (or expectancy) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff ("value"). Thus, it represents the average amount one "expects" to win per bet if bets with identical odds are repeated many times. Note that the value itself may not be expected in the general sense, it may be unlikely or even impossible.

The formula for expected value is:

(Average win x probability of win) - (average loss x probability of loss)

For example, an American roulette wheel has 38 equally possible outcomes. A bet placed on a single number pays 35-to-1 (this means that you are paid 35 times your bet and your bet is returned, so you receive 36 times your bet). So the expected value of the profit resulting from a $1 bet on a single number is, considering all 38 possible outcomes: $( 36 \times \frac{1}{38} )- (1 \times \frac{37}{38})$, which is about -$0.0263. Therefore one expects, on average, to lose over two cents for every dollar bet. Although the casino will occasionally pay out large sums to lucky customers, over time it enjoys a positive expectancy and thus is bound to make money.

Expected value is a key factor when testing the viability of trading systems.