Binomial Probability Calculator
Find the probability of getting exactly, at most or more than k successes across n independent trials with a fixed success probability.
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Calculation breakdown
- Binomial probability
- P(X = k) = nCk × p^k × (1 − p)^(n − k)
- Mean
- E(X) = n × p
Worked example
Flipping a fair coin 10 times, the probability of getting exactly 4 heads (p = 50%) is 10C4 × 0.5⁴ × 0.5⁶ ≈ 20.5%.
Assumptions
- Assumes each trial is independent with the same probability of success, and results are integer counts of successes.
- Not suitable for very large n (over a few hundred) due to floating-point rounding in the power terms.
How this calculator works
Enter the number of trials, the probability of success on each trial, and the number of successes you're interested in. The calculator applies the binomial probability formula to return P(X = k), P(X ≤ k) and P(X > k).
Frequently asked questions
What counts as a 'trial' here?
A trial is any repeated event with only two outcomes (success/failure) and the same success probability each time, such as a coin flip or a pass/fail test.
What does P(X ≤ k) mean?
It's the cumulative probability of getting k or fewer successes across all the trials, adding up the probabilities for 0 through k successes.
Does this work for very large numbers of trials?
It's reliable for typical classroom and business scenarios, but very large trial counts (several hundred or more) can lose precision due to floating-point rounding.