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

These two different geometric distributions should not be confused with each other. Often, the name shifted geometric distribution is adopted for the former one (distribution of ${\displaystyle X}$); however, to avoid ambiguity, it is considered wise to indicate which is intended, by mentioning the support explicitly.

The geometric distribution gives the probability that the first occurrence of success requires ${\displaystyle k}$ independent trials, each with success probability ${\displaystyle p}$. If the probability of success on each trial is ${\displaystyle p}$, then the probability that the ${\displaystyle k}$-th trial is the first success is

$${\displaystyle \Pr(X=k)=(1-p)^{k-1}p}$$

for ${\displaystyle k=1,2,3,4,\dots }$

The above form of the geometric distribution is used for modeling the number of trials up to and including the first success. By contrast, the following form of the geometric distribution is used for modeling the number of failures until the first success:

$${\displaystyle \Pr(Y=k)=\Pr(X=k+1)=(1-p)^{k}p}$$

for ${\displaystyle k=0,1,2,3,\dots }$