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

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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:

- Rolling 8 or more sixes from 20 die rolls (~1 in 89)
- Getting 35 or more heads from 50 coin tosses (~1 in 303)
- Getting 6 reds from 6 spins on a roulette wheel (~1.3%)
- A room of 50 people containing nobody born on a Sunday (~1 in 2,225)
- A room of 365 people containing nobody born on 1st January (~37%)

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)p^{k}(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.

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.