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) = 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.
Set this number aside for a moment.
Step 6: Work the third part of the formula.
q(n – X)
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.