Binomial Distribution Calculator

Binomial Distribution is the probability distribution of number of successes in n number of events (trials). For instance a die is thrown randomly 10 times, what's the probability of a five on rolling die in 7 out of 10 times? Here probability of a five (p) is (1/6) on a single throw as die has six sides. n is the the number of times die is thrown which is 10. k is the number of times we want success which is 7.


P(X < ) means probability of less than successes. Similarly P(X > ) refers to probability of more than successes.
How to calculate P(X < )
To calculate P(X < ), we need to sum all the probabilities from P(X = 0) through P(X = ). See the details below.

How to calculate P(X > )
To calculate P(X > ), we need to sum all the probabilities from P(X = ) through P(X = ). See the details below.

Binomial Distribution Table

The sum of all the probabilities shown below will be 1.

Important Points
  • Probability of success must lie between 0 and 1.
  • Number of trials can't be less than or equal to 0. It must be a whole number, can't be in decimals.
  • Number of successes can't be greater than the number of trials.

Binomial Distribution has the following properties -

  • Each trial has only two possible outcomes - success or failure.
  • The probability of a success on any trial would be same.
  • No trials depend on each other.

There is no specific definition of success. You can define it as a desired outcome. It depends on the problem statement. See the examples below -

  • If you flip a coin 10 times, what is the probability of getting more than four tails? Here "success" is getting tail.
  • Suppose you play a video game (let's say mario). What is the probability that you win 3 of the 10 times. Here "success" is winning the game.
  • Suppose school administation wants to know the number of students who take out their names from the randomly selected statistics class. The participation rate in the statistics class is 40% for any given school session. It means withdrawal rate is 60% i.e (1-0.4). Here "success" is a student who took out his/her name.
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