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# Expected Value of Winning

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

## Answers to Questions

### How to compute a mathematical expected value?

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, he 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.

### What is a fair game?

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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