Statistics How To

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

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

  1. Gina

    The step by step for binomial distribution was very helpful. However, I am running into a lot of problems on homework that read “atleast 1” or “3 or more” instead of just saying “exactly 3.” Is there anything on this website that might be able to explain that to me?

  2. jessica

    …thanks for this!!!it really helps me a lot…i can understand now,more that our teacher had taught us before..thank you so much..!!!

  3. joy

    10! / ((10 – 7)! × 7!) how to solve this? do u have the detailed solution of how u come up with 120 as a result?

  4. Andale


    What did you need to know? Post your question in the forum and one of our mods will help :)


  5. George

    10! / ((10 – 7)! × 7!) is as follows
    10! is 10x9x8x7x6x5x4x3x2x1 …. find that answer = 3628800. It is the answer for the top of the fraction.
    Then (10-7)! is the same as 3! which is 3x2x1 = 6
    Then 7! is 7x6x5x4x3x2x1 … find that answer = 5040
    The final answer is
    3628800/6×5040 = 3628800/30240 = 120
    Hope this helps.

  6. Andale

    1/ .6 is the given probability from the question. so .4 is just 1-.6
    2/ pX
    = .67
    = .0.0279936
    is the second part of the binomial formula (px)


  7. karim

    if heterozygous parents have 5 children what is the probability that 2 will be albino.

    please help i don’t understand how to solve this

  8. Sariey Summer

    How do you find the answer if the problem says fewer than 5? What if the problem says between 8 and 10 inclusively?

  9. Andale

    If you’ve got a TI89, you can use the binomcdf feature. Otherwise, you’ll have to repeat your calculations manually. For example, to calculate between 8 and 10 you’ll need to know P(8) + P(9)+P(10).

  10. 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

  11. 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