# Binomial Distribution

How to find the expected value, variance, and the standard deviation from the binomial distribution data: formulas, examples, and their solutions.

## Formula

For a binomial experiment

with a very large *n*:

E(*X*) = *np*

Expected value

V(*X*) = *npq*

σ(*X*) = √V(*X*)

= √*npq*

Variance, Standard deviation

## Example 1

To use the formulas,

write *n*, *p*, and *q*.*n* = 100 (100 trials)*p* = 1/2 (Probability of 'head' shown)*q* = 1 - 1/2 = 1/2 (Probability of 'head' not shown)

Probability of (not *A*, Complement)

E(*X*) = 100⋅(1/2)

V(*X*) = 100⋅(1/2)⋅(1/2)

You can find σ(*X*)

by directly square rooting the variance (√25)

or by using the formula (√100⋅(1/2)⋅(1/2)).

## Example 2

To use the formulas,

write *n*, *p*, and *q*.*n* = 360 (360 trials)*p* = 1/6 (Probability of '1' shown)*q* = 1 - 1/6 = 5/6 (Probability of '1' not shown)

Probability of (not *A*, Complement)

E(*X*) = 360⋅(1/6)

V(*X*) = 360⋅(1/6)⋅(5/6)

You can find σ(*X*)

by directly square rooting the variance (√50)

or by using the formula (√360⋅(1/6)⋅(5/6)).