The Mathematics of Gambling: What is “expected value”?

You may have heard this term before, especially if you have read some of our other “Mathematics of Gambling” pages, as we refer to it a little bit throughout our descriptions of other features of gambling maths. The concept of expected value is a key concept to understand if you wish to build a comprehensive understanding of the mathematics of gambling generally. Expected value, in a specific context of gambling, relates to how well or poorly a player can expect to do in a given casino over a series of fixed bets. A fixed bet is, for example, repeatedly betting $20 on red in roulette.
The Calculation of Expected Value
In order to calculate the expected value of a particular gambling scenario, you must use the following formula:
[(probability of winning) x (amount won per bet) + (probability of losing) x (amount lost per bet)]

This can be difficult to understand in the abstract, so let’s use an example to help out. Let us say that a player is playing roulette and would like to work out what the Expected Value of a certain bet – let’s say $10 on black – over a period of time will be.

The probability of winning a bet on black in roulette is 18 in 38 – this is easy to calulate as there are 38 pockets on the wheel (in American roulette anyway. In French or European Roulette there are only 37 pockets), and 18 of them are black. Therefore, this means that the probability of losing is 20 in 38. The amount won per bet and lost per bet is the same – that is, the amount of the bet, which is $10. So, let’s substitute these figures into the equation above:
(18/38) x 10 + (20/38) x (-10) = -0.526
This figure of -0.526 represents the fact that if a player makes a bet of $10 repetitively, he or she 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. If you want more information about the house edge, check our Frequently Asked Question What is theHouse Edge?

If you want to find out the expected value for a series of identical, then you will also need to have some knowledge of probability theory, seeing as probabilities are an important part of the equation that is used to figure out expected value. To get an in depth understanding of probability theory, you can check our Frequently Asked Question, What is probability theory? Or do some research of your own. We can provide you with a brief description here. Probabilites can be expressed as odds – for example, one in ten. They can also be expressed as a fraction, as in 1/10 or as a percentage – 10%, or as a decimal 0.1. The most common of these, and the easiest to work with mathematically, is the decimal. In order to find the probability of something, simply divide the total number of possibilities for a particular event occurring – such as landing on black in roulette – by the total number of all outcomes that exist in that same scenario – spinning the roulette wheel. In this example, we would divide 18 by 38 and find that the probability of landing on black in any given spin of a roulette wheel is around 0.474 or around 47%

Expected Value