Statistics How To

Binomial Distribution: Formula, What it is and How to use it

Binomial Theorem >


  1. What is a Binomial Distribution?
  2. The Bernoulli Distribution
  3. The Binomial Distribution Formula
  4. Worked Examples

What is a Binomial Distribution?

A binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. The binomial is a type of distribution that has two possible outcomes (the prefix “bi” means two, or twice). For example, a coin toss has only two possible outcomes: heads or tails and taking a test could have two possible outcomes: pass or fail.

what is a binomial distribution

A Binomial Distribution shows either (S)uccess or (F)ailure.

The first variable in the binomial formula, n, stands for the number of times the experiment is performed. The second variable, p, represents the probability of one specific outcome. For example, let’s suppose you wanted to know the probability of getting a 1 on a die roll. if you were to roll a die 20 times, the probability of rolling a one on any throw is 1/6. Roll twenty times and you have a binomial distribution of (n=20, p=1/6). SUCCESS would be “roll a one” and FAILURE would be “roll anything else.” If the outcome in question was the probability of the die landing on an even number, the binomial distribution would then become (n=20, p=1/2). That’s because your probability of throwing an even number is one half.

Binomial distributions must also meet the following three criteria:

  1. The number of observations or trials is fixed. In other words, you can only figure out the probability of something happening if you do it a certain number of times. This is common sense — if you toss a coin once, your probability of getting a tails is 50%. If you toss a coin a 20 times, your probability of getting a tails is very, very close to 100%.
  2. Each observation or trial is independent. In other words, none of your trials have an effect on the probability of the next trial.
  3. The probability of success (tails, heads, fail or pass) is exactly the same from one trial to another.

Once you know that your distribution is binomial, you can apply the binomial distribution formula to calculate the probability.

What is a Binomial Distribution? The Bernoulli Distribution.

The binomial distribution is closely related to the Bernoulli distribution. According to Washington State University, “If each Bernoulli trial is independent, then the number of successes in Bernoulli trails has a Binomial Distribution. On the other hand, the Bernoulli distribution is the Binomial distribution with n=1.”

A Bernouilli distribution is a set of Bernouilli trials. Each Bernouilli trial has one possible outcome, chosen from S, success, or F, failure. In each trial, the probability of success, P(S)=p, is the same. The probability of failure is just 1 minus the probability of success: P(F) = 1-p. (Remember that “1” is the total probability of an event occurring…probability is always between zero and 1). Finally, all Bernouilli trials are independent from each other and the probability of success doesn’t change from trial to trial, even if you have information about the other trials’ outcomes.

What is a Binomial Distribution? Real Life Examples

Many instances of binomial distributions can be found in real life. For example, if a new drug is introduced to cure a disease, it either cures the disease (it’s successful) or it doesn’t cure the disease (it’s a failure). If you purchase a lottery ticket, you’re either going to win money, or you aren’t. Basically, anything you can think of that can only be a success or a failure can be represented by a binomial distribution.

The Binomial Distribution Formula

Binomial Distribution formula

A Binomial Distribution shows either (S)uccess or (F)ailure.

The binomial distribution formula is:

b(x; n, P) = nCx * Px * (1 – P)n – x

b = binomial probability
x = total number of “successes” (pass or fail, heads or tails etc.)
P = probability of a success on an individual trial
n = number of trials

Note: The binomial distribution formula can also be written in a slightly different way, because nCx = n!/x!(n-x)! (this binomial distribution formula uses factorials (What is a factorial?). “q” in this formula is just the probability of failure (subtract your probability of success from 1).

Sample Problem Using the First Binomial Distribution Formula

Q. A coin is tossed 10 times. What is the probability of getting exactly 6 heads?

I’m going to use this formula: b(x; n, P) – nCx * Px * (1 – P)n – x
The number of trials (n) is 10
The odds of success (“tossing a heads”) is 0.5 (So 1-p = 0.5)
x = 6

P(x=6) = 10C6 * 0.5^6 * 0.5^4 = 210 * 0.015625 * 0.0625 = 0.205078125

Tip: You can use the combinations calculator to figure out the value for nCx.

How to Work a Binomial Distribution Formula: Example #2

The binomial distribution formula can calculate the probability of success for binomial distributions. Often you’ll be told to “plug in” the numbers to the formula and calculate. This is easy to say, but not so easy to do–unless you are very careful with order of operations, you won’t get the right answer. If you have a Ti-83 or Ti-89, the calculator can do much of the work for you. If not, here’s how to break down the problem into simple steps so you get the answer right–every time.


Step 1:: Read the question carefully.  Sample question: “80% of people who purchase pet insurance are women.  If 9 pet insurance owners are randomly selected, find the probability that exactly 6 are women.”

Step 2:: Identify ‘n’ and ‘X’ from the problem. Using our sample question, n (the number of randomly selected items) is 9,  and  X (the number you are asked to find the probability for) is 6.

Step 3: Work the first part of the formula. The first part of the formula is

n! / (n – X)!  X!

Substitute your variables:

9! / ((9 – 6)! × 6!)

Which equals 84. Set this number aside for a moment.

Step 4: Find p and q. p is the probability of success and q is the probability of failure. We are given p = 80%, or .8. So the probability of failure is 1 – .8 = .2 (20%).

Step 5: Work the second part of the formula.

= .86
= .262144

Set this number aside for a moment.

Step 6: Work the third part of the formula.

q(n – X)
= .2(9-6)
= .23
= .008

Step 7: Multiply your answer from step 3, 5, and 6 together.
84  × .262144 × .008 = 0.176.

That’s the simple explanation. For a more technical definition of the binomial and associated functions, check out the Wolfram page.

WSU. Retrieved Feb 15, 2016 from:

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Binomial Distribution: Formula, What it is and How to use it was last modified: October 15th, 2017 by Andale

28 thoughts on “Binomial Distribution: Formula, What it is and How to use it

  1. akindele ken

    good one! But pls, it would iron out complex examples such as prob of “altleast” combined with “or” & “and”.

  2. Jeffery rich

    Thank you so much. You simplified the formula where the probability of me remembering it is 100% success.

  3. sudarshan

    I have slightly different problem to solve and thru this example i am not able to decide n x p etc

    i have a 3695 balls of which 37 are red ,28 are blue 43 are yellow, When i take a sample of 100 , 1) what is the probability that NO red ball comes? and 2) ATLEAST 2 RED Balls OR ATLEAST 2 YELLOW balls will come? thank you

  4. sanddy

    pls solve for me

    An accountant is to audit 24 accounts of a firm. sixteen of these are of highly valued customer. if the accountant selects 4 of the accounts at roundom, what is the probability that he chooses at least one highly-valued account?

  5. Sandy

    The student body of a large university consists of 60% female students. A random sample of 8 students is selected. What is the probability that among the students in the sample at least 6 are female? Please solve it for me.

  6. Andale Post author

    Hi, Sandy,
    Just plug your numbers into the formula. Your p is .6, which would make your q .4.
    Note that because it’s “at least 6” you would have to work the formula three times, for n = 6,7, and 8.
    Then add all the probabilities up to get your answer.

  7. Jojo

    Why is the first part of the binomial probability formula the combination formula and not the permutation formula? Does order not matter? Why? For example, the probability that 3 out of 5 people have a blood type O. Does it not make a difference that persons 1, 3 and 5 have the O blood type rather than persons 1, 2, 3?

  8. sonam

    The mean and variance of a binomial distribution are 3 and 2 respectively. Find the probability that the number of successes are:
    1) at most 2
    2) at least 7.

  9. Andale Post author

    Where do you get stuck? i.e. what is your working out so far? Post it and I’ll be happy to take a look.