Expected Value

When building an understanding of the mathematics of gambling, a key concept to understand is that of Expected Value, which, in the gambling context, relates to how well a player can expect to do in a particular casino game over a series of fixed bets, for example repeatedly betting £10 on black in roulette.

Calculation of Expected Value

The expected value of a particular gambling scenario is worked out as follows:

[(probability of winning) x (amount won per bet) + (probability of losing) x (amount lost per bet)]

To give an example, let’s say a player is playing roulette and wants to work out what the Expected Value of a certain bet – say £10 on black – over a period of time will be. The probability of winning (see the Roulette Mathematics page for more info on Roulette probabilities) a bet on black is 18 in 38 and the probability of losing is therefore 20 in 38. The amount won per bet and lost per bet is the same (£10). If we substitute these figures into the equation above, we get:

(18/38) x 10 + (20/38) x (-10) = -0.526

This figure of -0.526 means that if a player makes a bet of £10 repetitively, he can theoretically expect to lose £.053 each time he makes a £10 bet. This loss is incurred because of the house edge that all casinos operate with, which ensures that they make a profit in the long-term.

Probability theory and Expected Value

To work out the Expected Value for a string of identical bets requires a knowledge of probability theory, since probabilities are an important part of the equation that is used to work out Expected Value. Probabilities can be expressed as odds (for example, one in five), or as a fraction (1/5) or as a percentage (20%) or as a decimal (0.2). The decimal version is the most mathematically common and is easiest to work with. To work out the probability of something, simply divide the total possibilities for a particular event occurring (such as landing on red in roulette) by the total number of outcomes that exist for a particular scenario (spinning the roulette wheel). So, in the example above, we would divide 18 by 38 and say that the probability of landing on red in any given spin of a roulette wheel is approximately 0.474, or around 47%.

Read more pages about casino game mathematics or browse our Gambling Mathematics Glossary.