Binomial Distribution Calculator - Exact, Cumulative & Tail Probabilities
Binomial Distribution Calculator
Probability of k successes in n trials, plus cumulative and tail probabilities
Every Kind of Probability
Get P(X = k), P(X ≤ k), P(X < k), P(X ≥ k), P(X > k) and, optionally, P(k ≤ X ≤ upper), all at once.
Accurate for Large n
Computed in log space and checked against exact integer arithmetic, so n = 10,000 works and tiny tail probabilities keep their precision.
See the Shape
A bar chart shows the distribution and highlights the outcomes you asked about.
Full Privacy
Everything is calculated in your browser.
When Does the Binomial Distribution Apply?
Use it when you repeat the same experiment a fixed number of times n, each trial has just two outcomes (success or failure), the probability of success p is the same every time, and the trials are independent. Counting heads in 10 coin flips, defective items in a batch, or people who answer yes in a survey are all binomial. The probability of exactly k successes is C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ.
The mean is n × p and the variance is n × p × (1 − p). "At least k" questions, such as the chance of 60 or more heads in 100 flips, use the upper tail P(X ≥ k), which is computed directly rather than as 1 minus a cumulative value, so even very small probabilities are accurate.
Key Takeaways
- Complete: Exact, cumulative, tail and range probabilities in one calculation.
- Validated: Rejects non-integer n or k, p outside 0 to 1, and k above n, with a clear message.
- Edge cases: p = 0 and p = 1 behave correctly.