What It Means and Examples

What Is the Multinomial Distribution?

The multinomial distribution is the type of probability distribution used in finance to determine the likelihood of a set of outcomes. The term describes calculating the outcomes of experiments involving independent events that have two or more possible defined outcomes.

The more widely known binomial distribution is a special type of multinomial distribution in which there are only two possible outcomes, such as true/false or heads/tails.

Understanding the Multinomial Distribution

The multinomial distribution applies to experiments in which the following conditions are true:

• Repeated: The experiment consists of repeated trials, such as rolling a die five times instead of just once.
• Independent: Each trial must be independent of the others. For example, if you roll two dice, the outcome of one die does not impact the outcome of the other die.
• Same probability: The probability of each outcome must be the same across each instance of the experiment. For example, if a fair, six-sided die is used, then there must be a one-in-six chance of each number being given on each roll.
• Specific outcome: Each trial must produce a specific outcome, such as a number between two and 12 if rolling two six-sided dice.

Staying with dice, suppose we run an experiment in which we roll two dice 500 times. Our goal is to calculate the probability that the experiment will produce the following results across the 500 trials:

• The outcome will be 2 in 15% of the trials;
• The outcome will be 5 in 12% of the trials;
• The outcome will be 7 in 17% of the trials; and
• The outcome will be 11 in 20% of the trials.

The multinomial distribution would allow us to calculate the probability that the above combination of outcomes will occur. Although these numbers were chosen arbitrarily, the same type of analysis can be performed for meaningful experiments in science, investing, and other areas.

Example of the Multinomial Distribution

In business, a financial analyst might use the multinomial distribution to calculate things such as the likelihood a company will report better-than-expected earnings while its competitors report disappointing earnings.

In investing, a portfolio manager might use the multinomial distribution to estimate the probability of:

• A small-cap index outperforming a large-cap index 70% of the time
• The large-cap index outperforming the small-cap index 25% of the time
• The indexes having the same (or approximate) return 5% of the time.

In this scenario, the trial might take place over a full year of trading days, using data from the market to gauge the results. If the probability of this set of outcomes is sufficiently high, the investor might be tempted to make an overweight investment in the small-cap index.

The Bottom Line

In finance, the multinomial distribution is a type of probability distribution used to describe the likelihood of a given set of outcomes. Unlike the binomial distribution, which has only two possible outcomes, the multinomial distribution describes two or more defined outcomes. The binomial distribution is a subtype of the multinomial distribution. In finance, the multinomial distribution can be used to estimate the probability of a set of occurrences and analyze the best course of action.