Liar's Poker Probabilities
The chances that the other players have at least the amount of a number you need to be able to call your bid when challenged, can be determined by the following two formulae:
Formula 1 P(at least X times C) = 1 - binomcdf (Y, 0.1, X-1)
With:
X = amount of the needed number
C = the needed number, which has a probability of 1/10 = 0.1
Y = the amount of unknown numbers, which is equal to 8 x amount of extra players
Example 1: you are playing a 2-player game and you want to determine whether the other player has at least 2 more sixes.
P(at least 2 times six) = 1 - binomcdf (8, 0.1, 1) = 0.18670...
So you have a chance of 18.69% that the other player has at least 2 sixes
Example 2: you are playing a 5-player game and you want to determine whether the other players have at least 4 more sevens.
P(at least 4 times seven) = 1 - binomcdf (32, 0.1, 3) = 0.3997...
So you have a chance of 39.97% that the other 4 players have at least 4 sevens.
Formula 2. In order to calculate the probability of at least X times C, you have to subtract each probability from X=1 till X=X-1 from 1.
P(X times C) = Y nCr X x 0.1X x 0.9Y-X
With:
X = amount of the needed number
C = the needed number, which has a probability of 1/10 = 0.1
Y = the amount of unknown numbers, which is equal to 8 x amount of extra players
Example: you are playing a 2-player game and you want to determine whether the other player has at least 2 more sixes.
P(at least 2 times six) = 1 - P(no six) - P(1 six)
P(no six) = 8nCr0 x 0.10 x 0.98 = 0.4305
P(1 six) = 8nCr1 x 0.11 x 0.97 = 0.3826
P(at least 2 times six) = 1 - 0.4305 - 0.3826 = 0.18670...
So you have a chance of 18.69% that the other player has at least 2 sixes
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Famous quotes containing the words liar and/or poker:
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—Oscar Wilde (1854–1900)
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—David Mamet (b. 1947)