Binomial Distribution

Martin McBride
Technological Singularity
14 min readApr 29, 2024


Suppose you rolled a fair dice, ten times. What is the probability that you would throw a six exactly three times?

This situation can be modelled by a binomial distribution so the result can be found using the probability mass function of the binomial distribution. The formula is stated here, but we will look at this in more depth in a little while:

In this article, we will find out what a binomial distribution is, and how to use the formula to solve problems like the one above. We will see where the formula comes from, and see some of the properties of the distribution.

Binomial distribution

The binomial distribution is a discrete probability distribution, which means it applies to a series of separate trials. In our example above, each roll of the dice counts as a trial. A trial is some kind of process that has two possible outcomes, succeed or fail. In our case, we say the trial succeeds if the dice comes up six, and fails if it is any other number.

In the formula above:

  • n is the total number of trials we will perform. In our case, we intend to roll the dice ten times.
  • k is the number of passes we are expecting. In our case, we are calculating the probability of exactly three passes (ie rolling a six on three occasions out of the ten attempts).
  • p is the probability of the success of each trial. In our case, success means rolling a six, so p is 1/6 with a fair dice.