The binomial distribution formula helps to check the probability of getting “x” successes in “n” independent trials of a binomial experiment. To recall, the binomial distribution is a type of probability distribution in statistics that has two possible outcomes. In probability theory, the binomial distribution comes with two parameters n and p.
The probability distribution becomes a binomial probability distribution when it meets the following requirements.
- Each trail can have only two outcomes or the outcomes that can be reduced to two outcomes. These outcomes can be either a success or a failure.
- The trails must be a fixed number.
- The outcome of each trial must be independent of each other.
- And the success of probability must remain the same for each trail.
Binomial Distribution Formula in Probability
The formula for the binomial probability distribution is as stated below:
| Binomial Distribution Formula | |
|---|---|
| Binomial Distribution | P(x) = nCr · pr (1 − p)n−r |
| Or, | P(x) = [n!/r!(n−r)!]· pr (1 − p)n−r |
Where,
- n = Total number of events
- r = Total number of successful events.
- p = Probability of success on a single trial.
- nCr = [n!/r!(n−r)]!
- 1 – p = Probability of failure.
Try This: Binomial Distribution Calculator
Solved Examples Using Binomial Distribution Formula
Question 1:
Toss a coin for 12 times. What is the probability of getting exactly 7 heads?
Solution:
Number of trails (n) = 12
Number of success (r) = 7
probability of single trail(p) = ½ = 0.5
nCr = [n!/r!] × (n–r)!
= 12!/ 7!(12 – 7)!
= 12!/ 7! 5!
= 95040120
= 792
pr = 0.57 = 0.0078125
To Find (1−p)n−r, calculate (1-p) and (n-r).
1 – p = 1 – 0.5 = 0.5
n – r = 12 – 7 = 5
(1−p)n−r = 0.57 = 0.03125
Solve P(X = r) = nCr. pr . (1−p)n−r
= 792 x 0.0078125 x 0.03125
= 0.193359375
The probability of getting exactly 7 heads is 0.19