Binomial probability calculator
Useful for questions like the chance of 5 heads in 10 coin tosses, or how many defective items turn up in a batch.
Loading the tool…
Processing and privacy: The result is calculated inside your browser and your data is not sent to any server.
How to use it
- Enter number of trials n, success probability p and number of successes k.
- The result appears straight away and updates as you edit.
- Copy the result, or copy a link that keeps the same values to share it.
How it is calculated
P(X = k) = C(n, k) × p^k × (1 - p)^(n - k). Cumulative values are sums of these terms. Mean = n × p, standard deviation = √(n × p × (1 - p)).
Example
Inputs: Number of trials n = 10, Success probability p = 0.5, Number of successes k = 5. Result: P(X = k): 0.24609375; P(X < k): 0.376953125; P(X ≤ k): 0.623046875.
Limits and caveats
- Assumes independent trials with the same success probability each time.
- Up to 100000 trials.
Questions about this tool
What is the chance of exactly 5 heads in 10 tosses?
C(10, 5) × 0.5^10 = 252 ÷ 1024 ≈ 0.2461, about 24.6%.
What is the difference between P(X ≤ k) and P(X < k)?
The first includes k itself and the second stops at k - 1. They differ by P(X = k).
When is the binomial model wrong?
When you draw without replacement from a small group, since the chance changes after each draw. Use the hypergeometric distribution instead.