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Roulette Expected Value & Probability Explained

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Expected value in roulette can be defined as the weighted average of how much you can win or lose in one full session. And, in order to calculate the expected value, players will need to know four things: the probability of winning, and the amount you’ll win if your bet gets drawn, as well as the probability of losing and the amount you’ll lose if you do.

You can insert actual amounts of money in your calculations if you’re calculating the expected value of one particular bet. Otherwise, general units can be used for calculations that cover bets of any size.

Because the American roulette wheel has 38 pockets of equal size, calculating the probability of the ball landing in either of the pockets is simple. Where the ball lands are based on chance, which means that the wheel has a uniform probability distribution.

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Roulette Table Probabilities

The probabilities in an American roulette game are as follows:

  • There are 38 pockets. The probability of the ball landing in either is 1/38.
  • There are 18 red pockets. The probability of the ball landing in either is 18/38.
  • There are 18 black pockets. The probability of the ball landing in either is 18/38.
  • There are 19 pockets that are either black or green. The probability of red not occurring is 19/38.
  • There are 19 pockets that are either red or green. The probability of red not occurring is 19/38.

The same kind of logic applies when calculating the probabilities found in European roulette. Instead of having 38 pockets, this variation only has 37 pockets, bringing the probability of the ball landing in either to 1/37.

How to Calculate Expected Value

In mathematics, the expected value represents the mean value of the outcomes. In this instance, it represents the mean value of the outcome of a bet in roulette. Below is a practical example of how you can calculate the expected value of any given game using American roulette probability values.

Expected Value Formula = ∑ [ x * P(x)]

Example: In roulette, a player can place a $5 bet on the number 17 and have a 1/38 probability of winning. If the ball lands on 17, the player wins $175, otherwise the casino takes the players $5. What is the expected value of the game to the player?

Calculate Expected Value

Therefore, a player who bets $5 in a single American roulette spin, can expect to lose 0.26 cents in that round. The same formula and logic can be used for calculating the expected value of any given game using European roulette probability values.

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