# Binomial Distribution

## What is the 'Binomial Distribution'

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

## BREAKING DOWN 'Binomial Distribution'

A binomial distribution summarizes the number of trials, or observations, when each trial has the same probability of attaining one particular value. The binomial distribution determines the probability of observing a specified number of successful outcomes in a specified number of trials. The expected value, or mean, of a binomial distribution is calculated by multiplying the number of trials by the probability of successes. For example, the expected value of the number of heads in 100 trials is 50, or (100 * 0.5).

## Bernoulli Trial

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 could only have two possible outcomes: success or failure. For example, 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 (the probability of flipping a head is 0.5) and the results of one trial do not influence the results of another.

## Binomial Distribution Example

The 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 number of trials. Then, multiply the product by the combination between the number of trials and the number of successes.

For example, assume that casino created a new game in which participants are able to 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 would be exactly six heads in 20 coin flips. The participant wants to calculate the probability of this occurring, and therefore, he uses the calculation for the 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 0.037, or 3.7%. The expected value was 10 heads in this case, wo the participant made a poor bet.