The binomial probability density function (aka pdf) is a function that models the likelihood of obtaining k successes from n trials where the likelihood of success on each trial is p.

Since p is the probability of success, the probability of failure is (1−p). We can then understand the binomial distribution using the binomial theorem:

Each term in the expansion is the probability of exactly k success:

Since k successes means n−k failures, we need to multiply the probabilities pk and (1−p)k. But since there are also many different ways of succeeding k times from n trials, we also need to multiply by nCr​.

On exams, binomial probabilities will be found using a calculator.

Example

A soccer player is taking shots on goal. On each attempt she has a 30% chance of scoring. What is the probability that she scores 4 goals in 12 attempts?

On a calculator, we find binompdf(12, 0.3, 4)=0.231.

Notice that this is the same as you get from the formula: