Casinos and Mathematical Expectation

11/04/11

In this article we will look into the concept of mathematical expectation as applied to casino gambling. Understanding the concept is important if you want to understand the optimum way to gamble at online and real casinos.

Mathematical expectation can be calculated by looking at every possible way to win and every possible way to lose, multiply each event by the probability of the event occurring, and taking the sum (adding and subtracting) of all the results.

It is easy to demonstrate this with a simple game such as tossing an unbiased coin. There is a 0.5 chance that it will come down heads and a 0.5 chance that it will come down tails. If we bet £1 that the next toss will be a head, then the mathematical expectation of the bet is:

If it comes down heads: 0.5 x 1 = 0.5
If it comes down tails: 0.5 x -1 = -0.5
Taking the sum: 0.5 + -0.5 = 0
So the mathematical expectation is 0.

If we change the odds to 3/2 then this would work out as follows:
If it comes down heads: 0.5 x 1.5 = 0.75
If it comes down tails: 0.5 x -1 = -0.5
Taking the sum: 0.5 + -0.5 = 0.25
So the mathematical expectation is 0.25

If we were to play the first game 1000 times, we would expect to win or lose very little, but with the second game we would anticipate winning approximately £250.
We can use this simple calculation to assess any potential bet in any game and it allows us to find the best game to play and the best decisions to make when playing that game. Mathematical expectation is mathematically very simple, but the results can be very powerful.