How to use it
- Type the Number of trials n, for example 10.
- Type the Success probability p as a percentage, for example 50%.
- Type the Number of successes k you want the probability for.
- Read P(X = k): exactly k successes, P(X ≤ k) and P(X ≥ k), then the Distribution table and How it is worked out.
The binomial formula
A binomial experiment has n independent trials, and each trial succeeds with the same probability p. The number of successes X then has the binomial distribution. The probability of exactly k successes is the number of ways to choose which k trials succeed, C(n, k), times the probability of one such pattern, pᵏ(1 − p)ⁿ⁻ᵏ.
The mean of the number of successes is np and its variance is np(1 − p). The standard deviation is the square root of the variance, so most of the results lie within about two standard deviations of the mean.
P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, with C(n, k) = n! ÷ (k! (n − k)!)
mean = np
variance = np(1 − p)Exactly, at most and at least
P(X ≤ k) adds the probabilities of 0, 1, ..., k successes, and P(X ≥ k) adds those of k, k + 1, ..., n. The two sides overlap at k, so P(X ≥ k) is also one minus P(X ≤ k − 1). The table lists each count with its probability and the cumulative probability up to it.
Assumptions and what the calculator does not cover
- The trials must be independent and have the same success probability. Drawing without replacement from a small group needs the hypergeometric distribution instead.
- Trials with more than two outcomes need the multinomial distribution; the total number of trials is limited to 1000 here.
- For the Poisson distribution of rare events with many trials, use the Poisson probability calculator.
Worked example: 10 trials with p = 50% and k = 3
C(10, 3) = 120, so P(X = 3) = 120 × 0.5³ × 0.5⁷ = 120 ÷ 1024 = 0.117188. The probability of at most 3 successes is 0.171875, and of at least 3 successes 0.945313. The mean is 5 and the variance 2.5.
| Successes j | P(X = j) | P(X ≤ j) |
|---|---|---|
| 0 | 0.000977 | 0.000977 |
| 1 | 0.009766 | 0.010742 |
| 2 | 0.043945 | 0.054688 |
| 3 | 0.117188 | 0.171875 |
Frequently asked questions
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the probability of exactly k successes. P(X ≤ k) adds up the probabilities of every count from 0 to k, so it is always at least as large.
When can I use the binomial distribution?
When you count successes in a fixed number of trials that are independent and have the same success probability, such as coin tosses, defective items in a batch or correct answers guessed at random.
How do I find the probability of at least one success?
Enter k = 1 and read P(X ≥ 1). It equals 1 minus the probability of no success, that is 1 − (1 − p)ⁿ.
Why is the variance np(1 − p)?
Each trial contributes a variance of p(1 − p), and the independent trials add up, which gives n times that value.
What happens when the probability is 0% or 100%?
The outcome is certain: with 0% the count is always 0, and with 100% it is always n. The calculator shows a probability of 1 for that count and 0 for the others.
Does the calculator send my numbers anywhere?
No. The calculation runs in your browser, and nothing you type is sent to our servers.
Sources
Figures and text last checked on Tuesday, November 10, 2026.