Binomial distribution calculator
Count successes in a fixed number of yes/no trials. The bars show every possible count; the ones you asked about light up.
Example: 10 multiple-choice questions with 4 options each, guessing: n = 10, p = 0.25.
P(X = 3) = C(10, 3) × 0.5^3 × 0.5^7 = 120 × 0.125 × 0.007813 = 0.1172
Full probability table
| k | P(X = k) | P(X ≤ k) |
|---|---|---|
| 0 | 0.0010 | 0.0010 |
| 1 | 0.0098 | 0.0107 |
| 2 | 0.0439 | 0.0547 |
| 3 | 0.1172 | 0.1719 |
| 4 | 0.2051 | 0.3770 |
| 5 | 0.2461 | 0.6230 |
| 6 | 0.2051 | 0.8281 |
| 7 | 0.1172 | 0.9453 |
| 8 | 0.0439 | 0.9893 |
| 9 | 0.0098 | 0.9990 |
| 10 | 0.0010 | 1.0000 |
When to use the binomial distribution
Use it when you count successes in a fixed number of independent yes/no trials that all have the same chance of success. A handy checklist is BINS: Binary outcomes, Independent trials, fixed Number of trials, Same probability each time.
P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
What each piece of the formula does
- pᵏ(1 − p)ⁿ⁻ᵏ is the probability of one particular sequence with k successes, such as SSFFF…
- C(n, k) counts how many different sequences have exactly k successes. This is the same path-counting that makes the Galton board produce a bell.
Worked example
A student guesses all 10 questions of a 4-option multiple-choice quiz. The chance of exactly 5 right is C(10, 5) × 0.25⁵ × 0.75⁵ = 252 × 0.000977 × 0.2373 ≈ 0.0584. The chance of passing with at least 6 is P(X ≥ 6) ≈ 0.0197, about 1 in 50.
Normal approximation
When np and n(1 − p) are both at least 10, the binomial is close to a normal distribution with mean np and SD √(np(1 − p)). Set n = 100 above and the bars trace out a bell. For exact answers, this calculator never needs the approximation.
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Questions students ask
What is the binomial probability formula?
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) = n! / (k!(n − k)!) counts the ways to arrange k successes among n trials.
When is a situation binomial?
When there is a fixed number of trials n, each trial has two outcomes, the probability of success p is the same every time, and the trials are independent (the BINS checklist).
What are the mean and standard deviation of a binomial distribution?
Mean = np and standard deviation = √(np(1 − p)). For 20 coin flips, the mean is 10 heads and the SD is about 2.24.
How do I calculate "at least" probabilities?
P(X ≥ k) = 1 − P(X ≤ k − 1). For "at least one", that is 1 − (1 − p)^n.