Liar's Poker - Liar's Poker Probabilities

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 nC_r 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) = 8nC_r0 x 0.10 x 0.98 = 0.4305
P(1 six) = 8nC_r1 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