Tool to compute an expected value for a game, the probability of winning indicates the chances of winning a given game, while expected value helps to know how much a player can earn (on average).

Expected Value of Winning - dCode

Tag(s) : Statistics

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The calculation of the mathematical expected value is to multiply the probability of winning by the bet multiplier (in case of winning).

Expected value is generally calculated for a bet of 1 unit. Multiply the probability to win by the bet value to know the expected gain.

__Example:__ The game of Casino's French Roulette with `37` boxes `0` to `36`. The player bets on `RED`. There are `18` red boxes (1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34 or 36) so `18` winning events and 19 losing events (0, 2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35). The probability of winning is `18/37`, the probability of losing is `19/37`, the bet multiplier is `2`. Expected gain for a bet of 1 is $$ \frac{35}{37}-\frac{36}{37} \approx -0.027 $$

So every time the player plays 1, he will lose on average 2.7% of his bet.

A fair game is a game in which all players have an equal chance of winning. The expected value is zero (equal to 0).

__Example:__ In the coin toss game, the player bets on TAILS, if he loses, he loses his bet, if he wins, he wins twice his bet.

There is one (1) winning event: the piece is returned on TAILS.

There are a total of two (2) events as possible: either the piece is on HEADS, or it is on TAILS.

Probability of winning: 1/2 = 50%

Expected value: (2-1) * 1 / 2-1 * 1/2 = 0

This game is fair.

The reasoning is the same for a die roll where a player will win 6 times his bet when he predicts the correct number.

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*Expected Value of Winning* on dCode.fr [online website], retrieved on 2024-06-15,

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