Binomial Distribution: Definition, Formula, Analysis, and Example

Binomial Distribution: Definition, Formula, Analysis, and Example

A 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.

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.

A binomial distribution is a standard discrete distribution used in statistics instead of a continuous one, such as a normal one. Given the number of trials in the data, this is because the binomial distribution only counts two states, typically represented as 1 (for a success) or 0 (for a failure). The binomial distribution thus represents the probability of x successes in n trials, given a success probability p for each trial.

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 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 number of heads in 100 trials of heads or tails is 50, or (100 × 0.5). Another typical 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 are 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.2 Bernoulli distribution is a particular case of binomial distribution where the number of trials n = 1.

Example of Binomial Distribution

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

For example, assume that a casino has created a new game where 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 precisely 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 ten 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 chance of getting six heads is on the left tail in red. You can see a lower chance of six heads occurring than seven, eight, nine, 10, 11, 12, or 13.