# 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.

- 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 trail must be independent of each other.
- 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.

n = Total number of events

r = Total number of successful events.

p = Probability of success on a single trial.

_{n}C_{r} = $\frac{n!}{r!(n − r)!}$

1 – p = Probability of failure.

### Solved Examples

**Toss a coin for 12 times. What is the probability of getting exactly 7 heads.**

**Question 1:**

**Solution:**

Number of trails (n) = 12

Number of success (r) = 7

probability of single trail(p) = $\frac{1}{2}$ = 0.5

Number of success (r) = 7

probability of single trail(p) = $\frac{1}{2}$ = 0.5

_{n}C_{r} = $\frac{n!}{r!(n – r)!}$

= $\frac{12!}{7!(12 – 7)!}$

= $\frac{12!}{7!5!}$

= $\frac{95040}{120}$

= 792

p^{r} = $0.5^{7}$ = 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 = 1-0.5 = 0.5

n-r = 12-7 = 5

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

Solve P(X = r) = _{n}C_{r. }p^{r }. $(1-p)^{n-r }$

= 792 x 0.0078125 x 0.03125

= 0.193359375

The probability of getting exactly 7 heads is 0.19