### DEFINITION of Multinomial Distribution

Multinomial distribution calculates the probability of more than two quantitatively specified outcomes occurring in a set of repeat trials or experiments. A multinomial distribution could show the results of tossing two dice, because each die can land on one of six possible values. By contrast, the results of a coin toss would be shown using a binomial distribution because there are only two possible results of each toss, heads or tails.

### BREAKING DOWN Multinomial Distribution

The validity of multinomial distributions rest on the assumptions that the repeated trials are independent (e.g., in the dice experiment, rolling a five does not have any impact on the number that will be rolled next); the probability of each possible result must be constant (on each roll, there is a one in six chance of any number on the die coming up); and each trial has a discrete outcome (each roll results in 2 through 12). Staying with dice, let's say in repeat rolling of two dice 500 times, you want to calculate the probability that 2 will come up 15% of the time, 5 will appear 12% of the time, 7 will be the number 17% of the time, and 11 will be the outcome 20% of the time. (26%, or 100% less the designated probabilities of these four numbers, will represent the balance of other combined numbers, but this is outside the bounds of the experiment.) The multinomial distribution formula will deliver a "p" value, or probability that the above combination of the four percentages will occur.

### Example in Investing

Multinomial distributions can be applied widely in science and engineering, but what about investing? A portfolio analyst might be interested in estimating the probability of (a) a small cap index outperforming a large cap index 70% of the time, (b) the large cap index outperforming the small cap index 25% of the time, and (c) the indexes having the same (or approximate) return 5% of the time on the first trading day of each week , given that each condition has equivalent probabilities (33.3%) of occurring. The trial takes place over an entire year. If the probability is great enough, the portfolio manager may be tempted to overweight small caps in the portfolio at the close of the trading day prior to the first trading day of the following week.