## What Is the Multinomial Distribution?

The multinomial distribution is the type of probability distribution used in finance to determine things such as the likelihood a company will report better-than-expected earnings while competitors report disappointing earnings. The term describes calculating the outcomes of experiments involving independent events which 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.

In finance, analysts use the multinomial distribution to estimate the probability of a given set of outcomes occurring.

### Key Takeaways

- The multinomial distribution is used in finance to estimate the probability of a given set of outcomes occurring, such as the likelihood a company will report better-than-expected earnings while its competitors report disappointing earnings.
- It's a probability distribution used in experiments with two or more variables.
- There are different kinds of multinomial distributions, including the binomial distribution, which involves experiments with only two variables.
- The multinomial distribution is widely used in science and finance to estimate the probability of a given set of outcomes occurring.

## Understanding Multinomial Distribution

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

- The experiment consists of repeated trials, such as rolling a die five times instead of just once.
- 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.
- 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.
- 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.

## Real-World Example of the Multinomial Distribution

In investing, a portfolio manager or financial analyst might use the multinomial distribution to estimate 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.

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.