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) = ^{n}C_{r · }p^{r }(1 − p)^{n−r} |

Or, | P(x) = [n!/r!(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}C_{r}= [n!/r!(n−r)]!- 1 – p = Probability of failure.

**Try This: **Binomial Distribution Calculator

### Solved Example Using Binomial Distribution Formula

** Question :**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

^{n}C_{r} = [n!/r!] × (n–r)!

= 12!/ 7!(12 – 7)!

= 12!/ 7! 5!

= 95040120

= 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)^{n−r} = 0.5^{5 }= 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.