In probability theory and statistics, the geometric distribution is either of two discrete probability distributions:

1. the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set , or
2. the probability distribution of the number Y = X-1 of failures before the first success, supported on the set .

Which of these one calls “the” geometric distribution is a matter of convention and convenience.

These two different geometric distributions should not be confused with each other. Often, the name shifted geometric distribution is adopted for the latter one (distribution of the number Y); however, to avoid ambiguity, it is considered wise to indicate which is intended, by mentioning the range explicitly.

We use the latter definition and its probability mass function is:

f(x,p) = (1-p)^(x-1)*p

support x = 0,1,2,3,4,5,…