Binomial Distribution Calculator
Enter trials, success probability and successes to get exact binomial probabilities.
What this tool does
The probability of exactly k successes in n independent trials, each with success probability p, is C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ. Getting exactly 4 heads in 10 fair coin flips has probability 210 ÷ 1,024 = 0.2051, and at most 4 heads is 0.3770. Enter your own n, p and k above.
Applies the binomial formula for a fixed number of independent yes-or-no trials.
Gives P(X = k) and the four cumulative probabilities, plus the mean and spread.
How to use it
- Enter the number of trials n.
- Enter the probability of success p, between 0 and 1.
- Enter the number of successes k.
- Read the probabilities and, for small n, the full table.
Why use this calculator
Gives the exact probability of k successes in n trials and all four cumulative forms (less than, at most, at least, more than) in one go. Binomial probabilities come up in quality control, surveys, medicine, games and statistics courses. The formula needs large factorials, and "at least" questions mean summing many terms, which is slow and error-prone by hand.
Why we built it
We built the binomial distribution calculator to give the exact probability and all four cumulative forms together, with the mean and spread. Count outcomes with the permutations and combinations calculator and combine events with the probability calculator.
Who this tool is for
- Statistics students and teachers
- Quality-control and survey analysts
- Anyone estimating the odds of repeated yes-or-no events
When to use it
- Finding the chance of at least k defects in a batch.
- Solving a statistics homework problem.
- Estimating the odds of a number of wins in repeated games.
- Checking a survey result against a claimed rate.
How it works
P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ, where C(n, k) = n! ÷ (k!(n − k)!). Cumulative probabilities add these terms: P(X ≤ k) sums from 0 to k, and P(X ≥ k) = 1 − P(X < k). The mean is np, the variance np(1 − p) and the standard deviation its square root. The calculator works with logarithms of factorials so large n does not overflow.
Example: n = 10, p = 0.5, k = 4. P(X = 4) = 210 × 0.5¹⁰ = 0.205078. P(X ≤ 4) = 386 ÷ 1,024 = 0.376953, P(X ≥ 4) = 0.828125, mean 5 and standard deviation 1.5811.
It assumes the trials are independent and p is the same each time. Sampling without replacement from a small population needs a hypergeometric model instead.
A worked example
These are the values the calculator opens with, so you can see exactly what it does with them. Change any value on the page and the result updates; the figures below come from the same calculation.
| Input | Value |
|---|---|
| Number of trials | 10 |
| Probability of success on each trial | 0.5 |
| Number of successes | 4 |
With those inputs, the calculator returns:
- P(X = 4): 0.205078
- P(X < k): 0.171875
- P(X ≤ k): 0.376953
- P(X ≥ k): 0.828125
- P(X > k): 0.623047
- Mean (n × p): 5
- Variance (n × p × (1 − p)): 2.5
- Standard deviation: 1.5811
What you give it, and what you get back
| Item | Details |
|---|---|
| n, p, k | Trials (1 to 5,000), probability of success and number of successes. |
| Output | P(X=k), P(X<k), P(X≤k), P(X≥k), P(X>k), mean, variance, standard deviation and a table for n up to 30. |
Issues you may run into
| What you see | Why it happens | What to do |
|---|---|---|
| A probability shows as 1.2e-7 | Very small probabilities are shown in scientific notation. | Read it as 0.00000012. |
| k is larger than n | You cannot have more successes than trials. | Reduce k or increase n. |
| My trials are not independent | The binomial model needs independence. | Use a different distribution for sampling without replacement. |
Assumptions and limitations
- Up to 5,000 trials.
- Independent trials with a constant probability.
Frequently asked questions
What is a binomial distribution?
The distribution of the number of successes in a fixed number of independent trials that each succeed with the same probability.
How do I calculate binomial probability?
Use C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, or enter n, p and k in the calculator.
What does at least k mean?
P(X ≥ k): the probability of k or more successes. It is one minus the probability of fewer than k.
What are the mean and standard deviation of a binomial?
The mean is n × p and the standard deviation is the square root of n × p × (1 − p).
When can I use a binomial distribution?
When there is a fixed number of trials, each trial has two outcomes, the probability is constant and the trials are independent.
Is my information sent anywhere?
No. The calculation runs in your browser and nothing leaves the page.