Binomial Distribution Calculator

For a binomial distribution with n trials and success probability p, compute the probability mass P(X=k), cumulative probability, upper tail, mean, and variance, with a bar chart of the distribution.

Input

For a binomial distribution with n trials and success probability p, compute the probability mass, cumulative probability, upper tail, mean, and variance at a given number of successes k.

Number of trials n

Enter an integer of 1 or more.

Success probability p

Enter a value between 0 and 1 (for example 0.5).

Number of successes k

Enter an integer from 0 to n.

Result

Probability mass P(X equals k = 3) with n = 10 and p = 0.5

0.1171875

Cumulative probability P(X at most k)

0.171875

Upper tail P(X at least k)

0.9453125

Binomial coefficient C(n, k)

120

Mean

Variance

2.5

Bar chart of the probability mass function

How it works

The probability mass is P(X=k) = C(n,k) p^k (1-p)^(n-k), where C(n,k) is the binomial coefficient.
The cumulative probability P(X at most k) is computed with the regularized incomplete beta function, so it stays stable even for large n.
The upper tail is P(X at least k) = 1 - P(X at most k-1).
The mean is np and the variance is np(1-p).
The binomial coefficient is computed with the log gamma function to avoid factorial overflow.
When the number of trials n is large, the bar chart focuses on the range around the mean.

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