## How to Solve a Binomial Distribution Problem Using the Binomial Formula

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 simple steps you should follow to get the answer right every time.  The ! symbol is a factorial; if you aren’t sure how to work factorials, click here.

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.

pX
= .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!

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## 24 Responses to “How to Solve a Binomial Distribution Problem Using the Binomial Formula”

1. ### How to Do Everything Statistics » How to read a binomial distribution table said:

Sep 25, 09 at 8:34 am

[...] How to use the binomial formula to solve a binomial problem. [...]

2. ### Mary Johnson said:

Sep 25, 09 at 2:51 pm

This helped me so much. It is starting to make sense. Thanks!

3. ### Angie Widdows said:

Oct 01, 09 at 10:05 am

To someone not studying the examples in MathZone, this is a great example. I really try to follow the examples I see on-line and this example mirror’s that. It was helpful.

4. ### Vanessa DuBarry said:

Oct 01, 09 at 10:54 am

TO ME THIS QUESTION IS REALLY DIFFICULT TO UNDERSTAND, WHEN I WAS DOING THIS PROBLEM ON MATHZONE I COULDNT FIGURE IT OUT , AND THE WORST PART WAS THAT WHEN YOU CLICK THE SHOW ME BUTTON IT DOESNT EXPLAIN ANYTHING, SO FOR ME TO BE ABLE TO HAVE THIS SORT OF EXPLANATIONS AND STEP BY STEP PROBLEMS REALLY HELP, I AM STILL HAVING A BIT OF A TROUBLE BECAUSE JUST BY LOOKING AT THE PROBLEM AND SEEING THAT EXCLAMATION MARK IN IT MAKES ME SO CONFUSED. WHAT DOES THAT REALLY MEAN?

5. ### Stephanie said:

Oct 04, 09 at 8:25 am

It’s a factorial.

Stephanie

6. ### Scott Selph said:

Oct 05, 09 at 6:42 am

This made for a very difficult problem solver. There was to much to keep up with. In my google search I was able to find a table and was able to make that work. But the binomial calculator is still much easier.

7. ### Stephanie said:

Oct 06, 09 at 2:29 am

Hi Scott,
There are some tables on this site (look up the top, under “tables”) but you are correct–the calculator is much easier!!
Stephanie

8. ### Gina said:

Oct 12, 10 at 9:07 pm

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?

9. ### jessica said:

Feb 06, 12 at 3:26 pm

…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..!!!

10. ### joy said:

Aug 22, 12 at 8:13 am

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

11. ### Andale said:

Aug 26, 12 at 7:24 am

I didn’t add the steps for that, but you should be able to put that into most calculators, as written. The ! is a factorial (see this article on factorials).

Regards,
Stephanie

12. ### Taphel said:

Sep 09, 12 at 12:44 pm

Please assist me on how to use the Binomial distribution formular

13. ### Andale said:

Sep 12, 12 at 7:24 am

Taphel,

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

Stephanie

14. ### Ekta singh said:

Sep 20, 12 at 5:22 am

It is perfect explanation for those who haven’t studied stats..it helped me alot.thanks

15. ### Andale said:

Sep 21, 12 at 4:17 pm

Stephanie

16. ### Iris said:

Sep 29, 12 at 8:36 pm

Very heplful. It all makes more sense. Thanks….

17. ### George said:

Oct 02, 12 at 12:19 pm

Joy,
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
3628800/6×5040 = 3628800/30240 = 120
Hope this helps.
George

18. ### karim said:

Oct 15, 12 at 11:18 am

thanks dears but why .6 and .4 (pX
= .67
= .0.0279936)

19. ### Andale said:

Oct 15, 12 at 11:37 am

Karim,
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)

Regards,
Stephanie

20. ### karim said:

Oct 15, 12 at 11:47 am

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

21. ### Sariey Summer said:

Oct 17, 12 at 4:08 pm

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

22. ### Andale said:

Nov 08, 12 at 10:25 am

Sarley,
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).

23. ### Hassan said:

Jan 27, 13 at 2:43 pm

you can also use excel formula under fx select Binomdist then you can use

24. ### Probability and Statistics Review for Applied Modeling – Joe Champion said:

Feb 27, 13 at 2:08 pm

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