The geometric distribution models players of a game 'in search of success' . When a player wins they stop playing. There are three conditions the game must satisfy:

1. When the player wins he stops playing, or at least the geometric distribution ceases to model the game at this point. If the player continues to play, a new geometric distribution is required.

2. The probability of winning each game is a constantIf the players get better as more games are played thenis not constant and the distribution cannot be geometric.

3. Each game is independent. If a player loses he is not more likely to win the next time and vice versa.

If the probability of winning isand the player wins – and stops playing – at the n^th attempt then the player must havefailures before this success. Since each failure is independent with probability the probability of this isand since they win on the next attempt with probability we havethis is called the probability mass function – discrete distributions have probability mass functions as opposed to the probability density functions for continuous distributions.

The expected number of attempts to win the game is

We can also find the variance:

Then

and

This is called the cumulative mass function.