Successes and Failures in a Run of Bernoulli Trials

Bernoulli trials are independent and their probability of success on each trial is a constant value, say . If is the number of successes in a run of Bernoulli trials, then is a binomial random variable with parameters and . If is the number of failures in the same sequence of trials, then is also a binomial random variable, but with parameters and . Any event expressed in terms of the number of successes can also be expressed in terms of the number of failures; for example, more than three successes is the same as fewer than failures.