About this calculator
You get the exact probability and every cumulative version at once.
How to use it
- Enter n, p and k.
- Read the probabilities.
The formula
P(X = k) = nCk × pᵏ × (1 − p)ⁿ⁻ᵏ
- n
- number of trials
- k
- number of successes
- p
- probability of success in one trial
Worked example
Exactly 6 heads in 10 tosses
- 10C6 × 0.5⁶ × 0.5⁴ = 210 ÷ 1,024 = 0.2051.
- P(X ≤ 6) = 0.8281.
What the result means
The mean is np and the standard deviation √(np(1 − p)).
Assumptions
- Trials are independent with the same p.
Limitations
- Sampling without replacement from a small group needs the hypergeometric distribution instead.
Frequently asked questions
What are the conditions for a binomial distribution?
A fixed number of trials, two outcomes each, the same probability each time, and independent trials.
What is the difference between P(X ≤ k) and P(X < k)?
P(X ≤ k) includes k itself; P(X < k) stops at k − 1.
When can I use the normal approximation?
When np and n(1 − p) are both at least about 10.
Last reviewed on 6 October 2026. Found a mistake? Tell us.