The binomial distribution describes the probability when a certain trial with only two possible outcomes is repeated ‘n’ number of times.

For example, suppose that a coin is tossed 10 times and we define X=number of times we get heads. Then X follows the binomial probability distribution.

**To find the mean of the binomial distribution** we use the formula,

- Mean µ = n*p where,

n = total number of trials and,

p = probability of success of a single trial.

**Example to find mean of binomial variable**: Suppose an archer tries to hit a mark 20 times. The probability of success in a single hit is 0.8. Find the average number of times the archer is able to hit the mark.

**Solution**: We are given n=20 and,

p= probability of a successful hit=0.9

Then Mean = n*p = 20*0.9 = 18 and so we conclude that the archer hits the target an average of 18 out of 20 times.

**To find the variance and standard deviation of the binomial distribution** we use the formula,

- Variance σ
^{2}= n*p*(1-p) - Standard Deviation σ = √{n*p*(1-p)} where,

n = total number of trials and,

p = probability of success of a single trial.

**Example to find variance and standard deviation of binomial variable**: Suppose a factory manufactures 1000 bulbs. The probability of a bulb being defective in a single hit is 10%. Find the mean and variance of the number of defective bulbs.

**Solution**: We are given n = 1000 and p=10% = 0.1

Then Mean = n*p = 1000*0.1 = 100 so on average we have a hundred defective bulbs.

Variance σ^{2}= n*p*(1-p) = 1000*0.1*0.9 = 90 and,

Standard Deviation σ = √{n*p*(1-p)} = √90 = 9.49