Expected Value in Roulette

Formula for expected value for both discrete and continuous random variables.
Formula for expected value. Discrete case is above and continuous case is below. C.K.Taylor

The concept of expected value can be used to analyze the casino game of roulette.  We can use this idea from probability to determine how much money, in the long run, we will lose by playing roulette. 

Background

A roulette wheel in the U.S. contains 38 equally sized spaces. The wheel is spun and a ball randomly lands in one of these spaces. Two spaces are green and have numbers 0 and 00 on them. The other spaces are numbered from 1 to 36.

Half of these remaining spaces are red and half of them are black. Different wagers can be made on where the ball will end up landing. A common bet is to choose a color, such as red, and wager that the ball will land on any of the 18 red spaces.

Probabilities for Roulette

Since the spaces are the same size, the ball is equally likely to land in any of the spaces.  This means that a roulette wheel involves a uniform probability distribution. The probabilities that we will need to calculate our expected value are as follows:

  • There are a total of 38 spaces, and so the probability that a ball lands on one particular space is 1/38.
  • There are 18 red spaces, and so the probability that red occurs is 18/38.
  • There are 20 spaces that are black or green, and so the probability that red does not occur is 20/38.

Random Variable

The net winnings on a roulette wager can be thought of as a discrete random variable.

If we bet $1 on red and red occurs, then we win our dollar back and another dollar. This results in net winnings of 1. If we bet $1 on red and green or black occurs, then we lose the dollar that we bet. This results in net winnings of -1.

The random variable X defined as the net winnings from betting on red in roulette will take the value of 1 with probability 18/38 and will take the value 1 with probability 20/38.

Calculation of Expected Value

We use the above information with the formula for expected value. Since we have a discrete random variable X for net winnings, the expected value of betting $1 on red in roulette is

P(Red) x (Value of X for Red) + P(Not Red) x (Value of X for Not Red) = 18/38 x 1 + 20/38 x (-1) = -0.053.

Interpretation of Results

It helps to remember the meaning of expected value to interpret the results of this calculation. The expected value is very much a measurement of the center or average. It indicates what will happen in the long run every time that we bet $1 on red.

While we might win several times in a row in the short term, in the long run we will lose over 5 cents on average each time that we play. The presence of the 0 and 00 spaces are just enough to give the house a slight advantage. This advantage is so small that it can be difficult to detect, but in the end the house always wins.