# Binomial Distribution Calculator

## Calculate probabilities of the binomial distribution step by step

The calculator will find the simple and cumulative probabilities, as well as the mean, variance, and standard deviation of the binomial distribution.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

### Your Input

Calculate the various values for the binomial distribution with $n = 20$, $p = 0.3 = \frac{3}{10}$, and $x = 5$.

### Answer

Mean: $\mu = n p = \left(20\right)\cdot \left(\frac{3}{10}\right) = 6$A.

Variance: $\sigma^{2} = n p \left(1 - p\right) = \left(20\right)\cdot \left(\frac{3}{10}\right)\cdot \left(1 - \frac{3}{10}\right) = \frac{21}{5} = 4.2$A.

Standard deviation: $\sigma = \sqrt{n p \left(1 - p\right)} = \sqrt{\left(20\right)\cdot \left(\frac{3}{10}\right)\cdot \left(1 - \frac{3}{10}\right)} = \frac{\sqrt{105}}{5}\approx 2.04939015319192.$A

$P{\left(X = 5 \right)}\approx 0.17886305056988$A

$P{\left(X \lt 5 \right)}\approx 0.237507778877602$A

$P{\left(X \leq 5 \right)}\approx 0.416370829447481$A

$P{\left(X \gt 5 \right)}\approx 0.583629170552519$A

$P{\left(X \geq 5 \right)}\approx 0.762492221122398$A