The binomial cumulative density function tells us the probability of obtaining k or fewer successes in n trials, each with a likelihood of success of p.

The formula, which you do not need to know or use, is the sum of the binomial pdf formula:

In practice, we use the binomcdf function on our calculators to determine these probabilities.

Example

A student is taking a 20 question multiple choice exam where each question is worth 1 point. The student needs to score 11 points for a 5, and 15 points for a 6.

Given that the probability the student answers each question correctly is 0.6, find the probability that he scored a 5.

Let X∼B(20,0.6) be the student's score. The student scores a 5 if 11≤X<15 ie 11≤X≤14. We can express this probability as the difference of two probabilities:

Using a calculator, we find

• P(X≤14)=binomcdf(20, 0.6, 14)=0.874401

• P(X≤10)=binomcdf(20, 0.6, 10)=0.244663

Subtracting we find P(11≤X≤14)=0.630.

Note that you get the same result from doing binompdf(20, 0.6, 11)

+binompdf(20, 0.6,12)

+binompdf(20, 0.6, 13)

+binompdf(20, 0.6, 14)