Binomial Distribution Calculator

The binomial distribution describes the chance of getting a certain number of “successes” in a fixed number of independent yes/no trials, where each trial has the same probability of success. This binomial distribution calculator returns the binomial probability for an exact count of successes, or the cumulative chance of getting at most or at least that many.

How to use this calculator

1. Enter the number of trials n — how many times the experiment is repeated (for example, 10 coin flips).
2. Enter the success probability p for a single trial, as a decimal between 0 and 1 (a fair coin is 0.5).
3. Enter the number of successes k you want the probability for.
4. Pick a mode: exactly P(X = k), at most P(X ≤ k), or at least P(X ≥ k). “At most” and “at least” add up several individual probabilities for you.

Worked example

Flip a fair coin 10 times. What is the probability of getting exactly 5 heads? Use n = 10, p = 0.5, k = 5, mode exactly. The binomial formula is:

P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)

Here C(n, k) is the number of ways to choose k successes from n trials (the “n choose k” count):

C(10, 5) = 252
P(X = 5) = 252 × 0.5^5 × 0.5^5
         = 252 × (1 / 1024)
         = 0.2461

So there is about a 24.6% chance of exactly 5 heads in 10 flips. For the formal definition and properties, see the NIST/SEMATECH e-Handbook on the binomial distribution.

Frequently asked questions

What is “n choose k” in the binomial formula?

C(n, k), read “n choose k”, counts how many different ways k successes can land among n trials, ignoring order. In the example above there are 252 distinct arrangements of 5 heads across 10 flips, and the binomial formula multiplies that count by the probability of any single arrangement.

When can I use the binomial distribution?

Use it when you have a fixed number of independent trials, each with only two outcomes (success or failure) and the same success probability p every time. Coin flips, pass/fail tests, and yes/no survey answers all fit this pattern.

What is the difference between “exactly”, “at most”, and “at least”?

Exactly gives the binomial probability for a single value of k. At most sums the probabilities from 0 up to k, and at least sums from k up to n, giving you cumulative chances instead of a single point.