Find the probability of exactly k successes in n independent trials, each with the same probability of success.
How it works
The binomial probability mass function is P(X=k) = C(n,k) × pᵏ × (1-p)ⁿ⁻ᵏ, where C(n,k) counts the number of ways to choose k successes from n trials. Ten coin flips (p=0.5) landing exactly 3 heads has probability 0.1172, about 11.7%.
What this does not include
This finds the probability of exactly k successes, not “k or fewer” or “k or more” (cumulative probability), which would need summing this formula across a range of k values.
How to use this calculator
- Enter the number of trials (n).
- Enter the number of successes you want the probability of (k).
- Enter the probability of success on a single trial (p).
A worked example
10 coin flips (p=0.5), probability of exactly 3 heads: 0.117188, about 11.7%.
5 trials at p=0.3, probability of exactly 2 successes: 0.3087, about 30.9%.
What the variables mean
| Variable | Meaning |
|---|---|
| n | Number of independent trials |
| k | Number of successes the probability is being found for |
| p | Probability of success on a single trial |
Edge cases worth knowing
k can’t exceed n — asking for the probability of more successes than there are trials has no meaningful answer, so the calculator declines to show a result.
This finds “exactly k,” not “k or more.” A cumulative probability (like “at least 3 heads”) would need summing this formula across every k value from 3 up to n, not a single evaluation.
What counts as a “trial” in the binomial distribution?
Any repeated, independent event with exactly two outcomes and the same success probability each time — coin flips, pass/fail tests, and yes/no survey responses are all classic examples.
Why does C(n,k) appear in the formula?
It counts how many different orderings of successes and failures produce exactly k successes — 3 heads in 10 flips can happen in many different sequences, and each is equally likely.
Is the coin-flip probability calculator on this site the same thing?
It’s the same formula with p fixed at 0.5 for a fair coin — this calculator generalizes it to any success probability, not just a coin flip.
