Binomial Distribution

What Is a Binomial Distribution?

A binomial distribution is a probability distribution that describes the number of successes in a fixed number of trials, where each trial has only two possible outcomes and the same chance of success. In finance, it is often used to model simple yes-or-no events, such as whether a stock finishes a period higher or lower, or whether a loan payment arrives on time.

Four conditions define the setting. The number of trials (n) is fixed in advance, each trial is independent of the others, each trial has exactly two outcomes (usually called success and failure), and the probability of success (p) stays the same from trial to trial. Each single trial is sometimes called a Bernoulli trial.

How It Works

The probability of getting exactly k successes in n trials is:

P(X = k) = (n choose k) × p^k × (1 − p)^(n − k)

Here, (n choose k) is the number of ways to arrange k successes among n trials, found using a combination formula. The term p^k is the chance of those k successes occurring, and (1 − p)^(n − k) is the chance of the remaining trials failing.

The distribution has two summary measures. The mean is n × p, the average number of successes expected over many repetitions. The variance is n × p × (1 − p), which measures how widely the results tend to spread.

Example

Suppose a fair coin is flipped 10 times. Each flip has a 0.5 chance of landing heads. There are 210 ways to get exactly 6 heads in 10 flips, so the probability is 210 × (0.5)^10, which equals 210/1024, or about 0.205. The expected number of heads is 5.

Advantages and Limitations

The binomial model is simple and transparent, and it gives exact probabilities for counting problems. Analysts use it in basic option pricing models, such as the binomial tree approach, and to reason about how many loans in a group might default, if defaults are assumed to be independent.

Its assumptions are the main limitation. Real markets rarely have a constant probability of success or fully independent outcomes. When the number of trials is large, the binomial distribution begins to resemble the normal distribution, which is often easier to work with.