Binomial Distribution Calculator
Enter the number of trials and the probability of success, then choose the event you want to measure—from exactly one count to a range or tail probability.
See the binomial probability
Enter your values, or try an example.
Your results and calculation will appear here.
Show probability table
| Successes (x) | P(X = x) | P(X ≤ x) |
|---|
Show calculation
How to use this calculator
- Enter the number of trials and success probability. Choose decimal or percent format.
- Choose exactly, at most, at least, between or outside and enter whole-number counts.
- Calculate. The PMF chart highlights outcomes included in your event.
Worked example: 10 coin flips
This example uses sample values, not your current inputs.
mode: exact · n: 10 · p: 0.5 · x: 5
Probability = 0.2461; Percentage = 24.6094%
The probability of exactly 5 successes is 24.6094%.
Example calculation
- n = 10; p = 0.5 per trial; selected event: P(X = 5).
- P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). C(n, k) counts ways to choose k successes from n trials.
- P(X = 5) = C(10, 5) × 0.5^5 × 0.5^5 ≈ 0.2461.
- C(10, 5) = 252; the probability factors are approximately 0.03125 and 0.03125. Multiplying the unrounded factors gives 0.2461.
- P(X = 5) ≈ 0.2461; percentage ≈ 24.6094%.
- Mean = np = 5; variance = np(1 − p) = 2.5; SD = 1.5811.
At least 7 heads in 10 flips
For 10 fair coin flips, enter n = 10, p = 0.5, mode = At least, and x = 7.
P(X ≥ 7) = 1 − P(X ≤ 6) = 1 − 0.828125 = 0.171875, or 17.1875%. “At least” includes 7, 8, 9 and 10 heads.
Formula & assumptions
P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ; mean = np; variance = np(1 − p).
- Trials are independent, with two outcomes each.
- Success probability is constant across trials.
- Counts are inclusive for at most, at least and between. Outside excludes both endpoints.
- Up to 10,000 trials. Charts display a central window for large distributions.
Binomial PMF vs. CDF
The binomial probability mass function (PMF) calculates the probability of exactly k successes, P(X = k). The cumulative distribution function (CDF) calculates the probability of k or fewer successes, P(X ≤ k). The cumulative calculation is also commonly called binomcdf.
For a binomial distribution with n trials and success probability p, the PMF uses the formula P(X = k) = C(n, k) × pk × (1 − p)(n − k). The CDF adds the probabilities from zero through k successes.
| Function | Probability |
|---|---|
| PMF: P(X = 2) | 0.2335 |
| CDF: P(X ≤ 2) | 0.3828 |
The PMF describes exactly two successes. The CDF includes zero, one, or two successes.
Sources
Reviewed: October 2026
Common questions
When can I use a binomial distribution?
Use it for a fixed number of independent trials, each with two outcomes and the same success probability.
Is binomial PDF the same as PMF?
No. A binomial distribution is discrete, so PMF is the correct term for the probability of an exact success count. PDF refers to probability density functions for continuous distributions. Some searches use "binomial PDF" informally, but the binomial PMF is the mathematically correct concept.
Probability reference table
Explore the Binomial Distribution Table to compare probabilities across parameter columns.
Use these tools for mathematical information, not as a substitute for checking assumptions. Read our methodology or disclaimer.