Binomial Distribution

What is Binomial distribution?

Binomial distribution is a probability distribution that is concerned with two possible outcomes.  It is the sum of all successes in repeated independent trials conducted on an identical population. This distribution requires all of the following conditions:

  • Each trial is independent of the others.
  • There are only two possible outcomes in a trial: success or failure.
  • A total number of n identical trials are performed.
  • The probability of success and failure does not change across trials.

What’s the Difference Between Binomial and Bernoulli Distributions? 

Binomial distribution is the sum of identically distributed and independent Bernoulli random variables.

For a classic example, if tossing a coin, the probability of landing on heads is p. The probability of getting tails is 1−p, with no other possible outcomes from the action. If you were betting on heads, your success after one coin toss would be called a Bernoulli random variable.

While each toss gives you 50/50 Bernoulli odds, over the course of 10 rounds you could win anywhere from 0 to 10 times. Your win-rate binomial distribution would be the sum of those trials, with parameters n and p.


The specific formula to calculate probability distribution using Binomial Distribution is: 

P (X = r ) = nCr (pˆr)* (1-p) * (n-r) 

Where: 
  • n : No. of trials 
  • r : No. of success 
  • p : the probability of success
  • 1 – p : Probability of failure
  • nCr : binomial coefficient given by n/k(n-k)