What Is Binomial Distribution?

Binomial distribution is a statistical distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters or assumptions. The underlying assumptions of binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive or independent of one another.

Understanding Binomial Distribution

To start, the “binomial” in binomial distribution means two terms—the number of successes and the number of attempts. Each is useless without the other.

Binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as normal distribution. This is because binomial distribution only counts two states, typically represented as 1 (for a success) or 0 (for a failure), given a number of trials in the data. Binomial distribution thus represents the probability for x successes in n trials, given a success probability p for each trial.

Binomial distribution summarizes the number of trials, or observations, when each trial has the same probability of attaining one particular value. Binomial distribution determines the probability of observing a specific number of successful outcomes in a specified number of trials.

Analyzing Binomial Distribution

A binomial distribution's expected value, or mean, is calculated by multiplying the number of trials (n) by the probability of successes (p), or n × p.

For example, the expected value of the number of heads in 100 trials of heads or tails is 50, or (100 × 0.5). Another common example of binomial distribution is estimating the chances of success for a free-throw shooter in basketball, where 1 = a basket made and 0 = a miss. The binomial distribution function is calculated as:

P_{( x : n , p )} = ⁿ C _x p ^x ( 1 - p ) ^{n - x}

Where:

• n is the number of trials (occurrences)
• x is the number of successful trials
• p is the probability of success in a single trial
• ⁿ C _x is the combination of n and x. A combination is the number of ways to choose a sample of x elements from a set of n distinct objects where order does not matter, and replacements are not allowed. Note that _nC_x = n! / r! ( n − r ) ! ), where ! is factorial (so, 4! = 4 × 3 × 2 × 1).

The mean of the binomial distribution is np, and the variance of the binomial distribution is np (1 − p). When p = 0.5, the distribution is symmetric around the mean—such as when flipping a coin because the chances of getting heads or tails is 50%, or 0.5. When p > 0.5, the distribution curve is skewed to the left. When p < 0.5, the distribution curve is skewed to the right.

The binomial distribution is the sum of a series of multiple independent and identically distributed Bernoulli trials. In a Bernoulli trial, the experiment is said to be random and can only have two possible outcomes: success or failure.

For instance, flipping a coin is considered to be a Bernoulli trial; each trial can only take one of two values (heads or tails), each success has the same probability, and the results of one trial do not influence the results of another. Bernoulli distribution is a special case of binomial distribution where the number of trials n = 1.

Example of Binomial Distribution

Binomial distribution is calculated by multiplying the probability of success raised to the power of the number of successes and the probability of failure raised to the power of the difference between the number of successes and the number of trials. Then, multiply the product by the combination of the number of trials and successes.

For example, assume that a casino created a new game in which participants can place bets on the number of heads or tails in a specified number of coin flips. Assume a participant wants to place a $10 bet that there will be exactly six heads in 20 coin flips. The participant wants to calculate the probability of this occurring, and therefore, they use the calculation for binomial distribution.

The probability was calculated as (20! / (6! × (20 - 6)!)) × (0.50)⁽⁶⁾ × (1 - 0.50)^{(20 - 6)}. Consequently, the probability of exactly six heads occurring in 20 coin flips is 0.0369, or 3.7%. The expected value was 10 heads in this case, so the participant made a poor bet. The graph below shows that the mean is 10 (the expected value), and the chances of getting six heads is on the left tail in red. You can see that there is less of a chance of six heads occurring than seven, eight, nine, 10, 11, 12, or 13 heads.

So how can this be used in finance? One example: Let’s say you’re a bank, a lender, that wants to know within three decimal places the likelihood of a particular borrower defaulting. What are the chances of so many borrowers defaulting that they would render the bank insolvent? Once you use the binomial distribution function to calculate that number, you have a better idea of how to price insurance and, ultimately, how much money to lend out and keep in reserve. 

The Bottom Line

The binomial distribution is an important statistical distribution that describes binary outcomes (such as the flip of a coin, a yes/no answer, or an on/off condition). Understanding its characteristics and functions is important for data analysis in various contexts that involve an outcome taking one of two independent values. It has applications in social science, finance, banking, insurance, and other areas. For instance, it can be used to estimate whether a borrower will default on a loan, whether an options contract will finish in-the-money or out-of-the-money, or whether a company will miss or beat earnings estimates.