5.4.2—Geometric distribution

If independent trials have constant success probability p, the number X of the first success has P(X=r)=(1−p)^{r−1}p for r=1,2,… .

Confirm that trials are independent and p is constant. “First success” counts trials including the successful one; a waiting-time variant may count failures instead.

With p=0.2, P(X=4)=0.8³×0.2=0.1024.

A geometric model does not describe the number of successes in a fixed number of trials; that is binomial.