Odds and mathematical expectation in roulette

Odds and mathematical expectation in roulette
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Odds and mathematical expectation in roulette
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Published: 24.12.2015 Updated: 21.05.2021
By Brad Isaac

Today we are going to talk about odds and mathematical expectation in roulette. Every serious gambler has to know some basic mathematical expectation. Without this knowledge you should not expect any success. Taking advantage of math can increase your chances.

Today we will have a look at the example of American roulette. This game has got 38 numbers (0, 00, 1-36) plus 11 different bets. Let’s say casino doesn’t cheat. The wheel is balanced and dealer doesn’t try to cheat you. According to the theory, this is an independent event. It means that outcome is unpredictable and it is not connected to the results of previous round. At the same time present outcome cannot affect upcoming rounds. In other words, if the ball stopped on black ten times in a row the next round can result in black or red with the same possibility.

Probability of hitting any number out of 38 is 37 to 1. Therefore if you place $1 on certain number your bet will win once in 38 spins. The casino will pay you in this case 35 to 1. It means that you will receive your buck and 35 additional dollars. In this case you will lose 2 dollars after 38 rounds. This is 5.26% of your total bets. Casino advantage is 5.26%.

In this table we have all variants of bets and winnings after 38 rounds:

With only one exception casino advantage is 5.26%. The exception is when you place bet on 5 numbers. In this case advantage of casino is 7.89%. You can evaluate these figures using another way. Formula of mathematical expectation is the following:

Mathematical expectation = [amount of favorable outcomes / amount of possible outcomes] x winning with favorable outcome + [amount of unfavorable outcomes / amount of possible outcomes] x bet

If you place $1 bet:

  • Bet on one number (Straight up) = (1/38) x 35 + (37/38) x (-1) = -2/38 = -0.0526 or (-5.26%)
  • Bet on five numbers (Five numbers) = (5/38) x 6 + (33/38) x (-1) = -3/38 = -0.0789 or (-7.89%)
  • Bet on red – eighteen numbers (Red bet) = (18/38) x 1 + (20/38) x (-1) = -2/38 = -0.0526 or (-5.26%)

The reason of negative result is that we have evaluated mathematical expectation for a player who places bet. You can just change sign and you will find out advantage of casino.

What do these figures mean? It means that casino expects to receive 5 cents profit from every placed dollar. If you think that this is quite tiny, just have a look at any huge casino. Turnover of a successful casino is enormous. And don’t be surprised that croupier tries to conduct as many rounds as possible during certain period of time.


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