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Expected Value of Winning

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 you can earn (on average).

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Expected Value of Winning -

Tag(s) : Mathematics, Combinatorics

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# Expected Value of Winning

## Expected Value Calculator

### I want to calculate

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 you can earn (on average).

### 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. Just multiply expected value by the bet to know the expected gain.

Consider the game of 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, when you wins, your bet is multiplied by 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 stake..

### 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).

Consider 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.