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
.
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