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.

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.

p^{X}

= .6^{7}

= .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)}

= .4^{3}

= .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*.

*Facebook page*and I'll do my best to help!

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

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

Hello, Annie,

The video will walk you through it. Your basically plugging in your info into the equation and solving. You have p = .01, 1 – p = 1 = .01 = .99 and n = 100. Once you’ve got the hang of the binomial distribution, you can find the Poisson distribution.

helpful in evry snse

How to find 0.9 raise to 20 without using calculator.Pls explain

You can solve the binomial formula by hand (see the video). A calculator isn’t needed :)