MATH · BINOMIAL PROBABILITY
Binomial Probability Calculator
Calculate P(X=k), P(X≤k), or P(X≥k) for a binomial distribution B(n,p) using the exact PMF. Includes a PMF bar chart for up to 50 trials.
| k | P(X = k) |
|---|---|
| 0 | 0.000977 |
| 1 | 0.009766 |
| 2 | 0.043945 |
| 3 | 0.117188 |
| 4 | 0.205078 |
| 5 | 0.246094 |
| 6 | 0.205078 |
| 7 | 0.117188 |
| 8 | 0.043945 |
| 9 | 0.009766 |
| 10 | 0.000977 |
About This Calculator
Compute exact binomial probabilities for any number of trials n and successes k with success probability p. Choose to compute P(X = k), P(X ≤ k), or P(X ≥ k). All three values are always displayed, and a PMF bar chart shows the full distribution shape.
How It Works
The binomial probability mass function gives the probability of exactly k successes in n independent Bernoulli trials, each with probability p of success. P(X ≤ k) sums the PMF from 0 to k; P(X ≥ k) sums from k to n. No critical-value tables are needed — all probabilities are computed exactly from the binomial formula.
The Formula
P(X=k) = C(n,k) × pᵏ × (1−p)^(n−k)
- n
- number of trials
- k
- number of successes
- p
- probability of success on each trial
- C(n,k)
- n choose k — the number of ways to arrange k successes
Frequently Asked Questions
- What is the binomial distribution?
- The binomial distribution B(n,p) models the number of successes in n independent yes/no trials, each with the same probability p of success. Classic examples include coin flips, quality-control pass/fail checks, and survey yes/no responses.
- What is P(X=k) versus P(X≤k)?
- P(X=k) is the probability of exactly k successes. P(X≤k) is the cumulative probability of k or fewer successes (the CDF). P(X≥k) is the probability of k or more successes (the survival function).
- What does the PMF bar chart show?
- The bar chart shows the probability of each possible outcome from 0 to n (or up to 50 outcomes if n > 50). Your chosen k is highlighted. The most likely outcome sits at the tallest bar, which occurs near the mean n×p.
- How are C(n,k) values computed for large n?
- For n > 170, standard JavaScript multiplication would overflow. The calculator uses BigInt arithmetic to compute C(n,k) exactly, then converts back to a floating-point probability, so results remain accurate even for large n.