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Binomial Formula: How to Solve a Binomial Problem

Binomial Theorem > Binomial Formula

Binomial Formula: Overview

A binomial is just an experiment that can have two outcomes: success or failure. For example, a coin toss can be a binomial experiment. If you wanted to know how many heads would come up if you tossed the coin three times, the “success” would be getting a heads on a single toss.

binomial formula

A coin toss can be a binomial experiment.

Although the concept of a binomial is simple, the binomial  formula–used to calculate the probability of success for binomial distributions–isn’t the easiest of formulas to work . If you use a TI-83 or TI-89, much of the work is done for you. However, if you don’t own one, here are the steps you should follow to get the answer right every time.  
Note: The ! symbol is a factorial (What is a factorial?)


Sample question: “60% of people who purchase sports cars are men.  If 10 sports car owners are randomly selected, find the probability that exactly 7 are men.”

Step 1:: Identify ‘n’ and ‘X’ from the problem. Using our sample question, n (the number of randomly selected items — in this case, sports car owners are randomly selected) is 10,  and  X (the number you are asked to “find the probability” for) is 7.

Step 2: Figure out the first part of the formula, which is:

n! / (n – X)!  X!

Substituting the variables:

10! / ((10 – 7)! × 7!)

Which equals 120. Set this number aside for a moment.

Step 3: Find “p” the probability of success and “q” the probability of failure. We are given p = 60%, or .6. therefore, the probability of failure is 1 – .6 = .4 (40%).

Step 4: Work the next part of the formula.

= .67
= .0.0279936

Set this number aside while you work the third part of the formula.

Step 5: Work the third part of the formula.

q(.4 – 7)
= .4(10-7)
= .43
= .0.064

Step 6: Multiply the three answers from steps 2, 4 and 5 together.
120  × 0.0279936 × 0.064 = 0.215.

That’s it!

If you’re interested in a more technical rundown on the binomial formula, check out this article.


If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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Binomial Formula: How to Solve a Binomial Problem was last modified: October 12th, 2017 by Stephanie

6 thoughts on “Binomial Formula: How to Solve a Binomial Problem

  1. Jonathan Ayoo

    My first encounter with biononial probability distribution wasnt easy I never understood anything in class but with this step by step solving its very clear to me

  2. Annie

    If the probability of obtaining a correct answer is p=0.01 what is the probability that out of set of 100 questions more than 2 will be wrong find using binomial distribution and poissions distribution separately
    How can I solve please help me