# Binomial probability calculator

The probability that a single trial is a success, e.g. 0.25, 1/4, 25%.
The number of successful trials, between 0 and n.
The total number of trials conducted, between 0 and .
P(X ≤ k) =
k successes or fewer from n trials.
P(X = k) =
Exactly k successes from n trials.
P(X ≥ k) =
k successes or more from n trials.
Format

This is a binomial probability calculator. Given that a single Bernoulli trial has a success probability of p, it tells you the probability of getting k successes out of n trials.

For example, it can tell you the probabilities of these events:

### Maths

The probability of exactly k successes from n trials, where each trial has a success probability of 0 ≤ p ≤ 1, is given by the formula:

P(X = k) = nCr(n, k)pk(1−p)n−k

where the combination function nCr(n, k) is given by:

nCr(n, k) =
n!
k!(n − k)!

The cumulative probabilities of obtaining at most or at least x successes is then obtained by summing the probabilities for 0 ≤ k ≤ x and for x ≤ k ≤ n respectively.

### Technical waffle

This tool uses a special kind of floating-point number class with a large exponent range. This means it can calculate very low probabilities but only to the same level of precision as an ordinary 64-bit float. The tool shows you 10 digits.