What is Binomial Distribution (With Formula)?

A binomial distribution is one of the important discreet probability distributions in statistics. In this article we will learn about binomial distributions.

Before we start, make sure that you know about the concept of combinations in probability and the formula for combinations. Learn about combinations formula. 

Understanding Binomial Distribution

If you flip a coin 100 times, what’s the probability the it will turn a Tails at least 60 times. We can solve this problem using the binomial distribution. Let’s take another example. You have a pool of stocks having returns either above 5% or below 5%. The probability of selecting a stock with above 5% returns is 0.70. You are going to pick up 5 stocks. What is the probability of picking 2 stocks with above 5% returns? From a probability perspective, both these are similar scenarios. We can use the binomial distribution to calculate this probability.
The binomial distribution has the following characteristics:

  • For each trial there are only two possible outcomes, success or failure.
  • Probability of success, p, of each trial is fixed.
  • There are n trials.
  • Each trial is independent
  • The binomial probability function defines the probability of x successes from n trials. 

Binomial Distribution Formula

Suppose there are n independent trials. Independent here means that knowing the outcome of one trial gives us no information about the outcome of the other trial. Each trial can only result in one of the two possible mutually exclusive outcomes. We can label these outcomes as success or failure. We call the probability of a success in one trial as p. Then the probability of failure will be 1 minus the probability of success, i.e, (1-p).

Let X represent the count of the number of successes in n trials. Then X has a probability distribution. The probability that X takes the value little x, represented as p(x) is the probability x successes in n trials. As in our example, the probability of 60 Tails (successes) in 100 trials or the probability of picking 2 stocks with above 5% returns (x) in 5 stocks picked (n).  x can take on any value between 0 and n.

The binomial probability function is given using the following formula.


Let’s take another example. We roll a dice with 6 sides 3 times. What is the probability that it will show 5 exactly 2 times? We can calculate this probability using the above binomial probability function.

Any given roll or trial has 6 possibilities. For us a success is to roll a 5 and a failure is rolling anything other than 5. We now have just 2 outcomes on any given trial. The probability of a success in an independent trial is 1/6. If X represents the number of 5 in 3 rolls, then X has a binomial distribution with n=3, and p=1/6. We are looking for the probability of getting a 5 exactly 2 times in 3 rolls, i.e., x=2. We can put this into our binomial formula.

p(2) = (3!/(3-2)!*2!)*(1/6)^2*(1-1/6)^(3-2)

p(2) = 0.0694

So, the probability of getting 5 exactly 2 times in 3 rolls is 0.0694

Check the Binomial Probability Distribution Table