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.

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


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