The Trinomial Distribution
A binomial random variable models the number of successes in trials, where the trials are independent and the only options on each trial are success and failure. A generalization of this called a multinomial distribution can be obtained by allowing more than two possibilities on each trial. When there are three possibilities on each trial, call them "perfect", "acceptable", and "failing", the result is a trinomial random variable. Letting be the number of perfects and the number of acceptables in trials, the image is a rendering of the joint probability mass function of and . The cuboid whose lower-left corner is at has height equal to the probability of perfects and acceptables in trials. Note that if is the probability of a trial being perfect and the probability of a trial being acceptable, then the probability of failure on the trial is .