## The Binomial Distribution Formula: Binomial Distribution Overview

The binomial distribution is a type of distribution in statistics 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 (for more information, see What is a Binomial distribution?).

Binomial distributions must also meet the following three criteria:

**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%.**Each observation or trial is**independent. In other words, none of your trials have an effect on the probability of the next trial.- 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.

## The Binomial Distribution Formula

The binomial distribution formula is:

**b(x; n, P) = _{n}C_{x} * P^{x} * (1 – P)^{n – x}**

Where:

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 _{n}C_{x} = 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) – _{n}C_{x} * P^{x} * (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) = _{10}C_{6} * 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 _{n}C_{x}.

## 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.

p^{X}

= .8^{6}

= .262144

Set this number aside for a moment.

**Step 6:** Work the third part of the formula.

q^{(n – X)}

= .2^{(9-6)}

= .2^{3}

= .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.

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

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

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Hello, Nazir,

Please post on our forum; one of our mods will be happy to help!

Statistics Help Forum

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

Hello Sudarshan,

Could you please post this on our forum?

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