The Binomial Probability distribution is an experiment that possess the following properties:

- There are fixed number of trials which is denoted by n.
- All the trials are independent.
- The outcome of each trial can either be a "success" or "failure".

- The probability of success remains constant and is denoted by p.

n = Total number of trials

x = Total number of successful trials

p = probability of success in a single trial

q = probability of failure in a single trial = 1-p

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Below are the problems for binomial probability distribution:

### Solved Examples

**Question 1: **A fair coin is flipped 6 times. What is the probability of getting exactly 2 tails ?

** Solution: **

n = total number of trials = 6

x = total number of successful trials = 3

p = probability of success in one trial = $\frac{1}{2}$

q = probability of failure in one trial = 1 - $\frac{1}{2}$ = $\frac{1}{2}$

P(x) = $\left ( \frac{6!}{2!\times 4!} \right )$ $\times$ $\left(\frac{1}{2}\right)^{3}$ $\times$ $\left(\frac{1}{2}\right)^{3}$

= $\frac{15}{64}$

= 0.234

**Question 2: **A die is rolled 5 times. Find the probability of getting exactly 2 fours ?

** Solution: **

Here

n = 5

x = 2

p = $\frac{1}{6}$

q = 1 - $\frac{1}{6}$ = $\frac{5}{6}$

Then

P(x) = $\left(\frac{5!}{2! \times 3!}\right)$ $\times$ $\left(\frac{1}{6}\right)^{2}$ $\times$ $\left(\frac{5}{6}\right)^{3}$

= 0.161

n = total number of trials = 6

x = total number of successful trials = 3

p = probability of success in one trial = $\frac{1}{2}$

q = probability of failure in one trial = 1 - $\frac{1}{2}$ = $\frac{1}{2}$

P(x) = $\left ( \frac{6!}{2!\times 4!} \right )$ $\times$ $\left(\frac{1}{2}\right)^{3}$ $\times$ $\left(\frac{1}{2}\right)^{3}$

= $\frac{15}{64}$

= 0.234

Here

n = 5

x = 2

p = $\frac{1}{6}$

q = 1 - $\frac{1}{6}$ = $\frac{5}{6}$

Then

P(x) = $\left(\frac{5!}{2! \times 3!}\right)$ $\times$ $\left(\frac{1}{6}\right)^{2}$ $\times$ $\left(\frac{5}{6}\right)^{3}$

= 0.161