Consider the case of Bernoulli trials where each individual event has a binary outcome ( = {0,1}). We define b(r; n, p) as the probability of getting r successes in n independent trials, each with independent trial success probability p. The binomial probability law tells us that

where

Note the following, achieved by summing over all possibilities.

The probability of r or fewer successes (the binomial distribution function) is

The probability of r or more successes is

The probability of more than r, but less than j, successes is