Binomial Distribution Calculator

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