How to calculate binomial probability
- Enter the number of trials n, the probability of success p on each trial, and the number of successes k.
- Read P(X = k) and the cumulative probabilities: at most k, fewer than k, at least k and more than k.
- The mean (np) and standard deviation (√(np(1 − p))) describe the whole distribution.
When to use the binomial distribution
The binomial distribution applies when you repeat the same yes/no experiment a fixed number of times, the trials are independent, and the success probability stays the same: coin flips, free throws, defective items in a batch, survey responses or patients responding to a treatment. The probability of exactly k successes is C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ.
For example, flipping a fair coin 10 times, the chance of exactly 5 heads is about 24.6%, and the chance of at least 8 heads is about 5.5%. Cumulative values are found by adding the individual probabilities. For large n, the calculation uses logarithms to stay accurate. When n is large and p is not extreme, the binomial is close to a normal distribution with the same mean and standard deviation. Everything runs in your browser.
FAQ
What is the probability of 5 heads in 10 flips?
About 0.2461, or 24.6%.
What is the difference between "at least" and "more than"?
At least k includes k; more than k does not.
Can p be a percentage?
Yes. Type 30% or 0.3.