For the negative binomial distribution the random variable is the number of failures before the rth success is observed. The distribution has two parameters. The parameter p is the probability of success on any one trial and the parameter r is the number of successes to be observed before the experiment is complete.

The geometric distribution is a special case with r equal to 1.

Example: : In the game of craps, you decide to play until you lose 5 games. You wonder how many games you will play with this termination rule. The probability of losing any one game is 0.5071. The games are a series of independent Bernoulli trials, and the random variable is the number of wins until the fifth loss. This is a situation described by the negative binomial distribution.

For the example, we perversely defined success as a “loss” with p the probability of a success equal to 0.5071 for the example. The random variable is the number of trials that result in 0 before the rth 1 is observed. For this case, r = 5.

It is important to remember that the random variable is not the total number of trials but the number of failed trials before the rth success. In the solution, the entry for 0 describes the probability that the first five plays were losses and there were no wins.