# Binomial Distribution Formula

The binomial distribution is a type of probability distribution in statistics that has two possible outcomes. In probability theory, the binomial distribution come with two parameters n and p.

The probability distribution becomes a binomial probability distribution when it meets the following requirements.

1. 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.
2. The trails must be a fixed number.
3. The outcome of each trail must be independent of each other.
4. And the success of probability must remain the same for each trail.

The formula for binomial probability is as stated below:

$\large P(x) = \frac{n!}{r!(n-r)!} . p^{r}(1-p)^{n} = C(n, r).p^{r}(1-p)^{n-r}$
Where,
n = Total number of events
r = Total number of successful events.
p = Probability of success on a single trial.

nCr = $\frac{n!}{r!(n − r)!}$

1 – p = Probability of failure.

### Solved Examples

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) = $\frac{1}{2}$ = 0.5

nCr = $\frac{n!}{r!(n – r)!}$
= $\frac{12!}{7!(12 – 7)!}$
= $\frac{12!}{7!5!}$
= $\frac{95040}{120}$
= 792

pr = $0.5^{7}$ = 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
$(1-p)^{n-r }$ = $0.5^{7}$ = 0.03125

Solve P(X = r) = nCr. p. $(1-p)^{n-r }$
= 792 x 0.0078125 x 0.03125
= 0.193359375

The probability of getting exactly 7 heads is 0.19