Binomial Probability Calculator
(Exactly, At Most, At Least and Between)
Enter the number of trials, the probability of success and the number of successes. The calculator shows every binomial probability for that k at once.
Between 0 and 1, for example 0.5
Optional. Gives P(k ≤ X ≤ this number)
Binomial Probability
💡 Formula Guide
This binomial probability calculator gives the chance of exactly k successes in n independent trials, and the cumulative chances of at most, less than, at least and more than k, all at once. It also gives the probability of a range and the distribution's mean, variance and standard deviation.
The Binomial Formula
For 6 heads in 10 flips of a fair coin, C(10, 6) = 210 and 0.5^10 = 1/1,024, so P(X = 6) = 210 ÷ 1,024 = 0.2051.
| Question | Notation | 10 flips, k = 6 |
|---|---|---|
| Exactly 6 | P(X = 6) | 0.205078 |
| At most 6 | P(X ≤ 6) | 0.828125 |
| Fewer than 6 | P(X < 6) | 0.623047 |
| At least 6 | P(X ≥ 6) | 0.376953 |
| More than 6 | P(X > 6) | 0.171875 |
| Between 6 and 8 | P(6 ≤ X ≤ 8) | 0.366211 |
Each cumulative answer is a sum of exact probabilities. At least 6 is P(6) + P(7) + P(8) + P(9) + P(10) = (210 + 120 + 45 + 10 + 1) ÷ 1,024.
A Case Where p Is Not One Half
The chance of exactly 2 sixes in 10 rolls of a die is C(10, 2) × (1/6)^2 × (5/6)^8 = 45 × 0.02778 × 0.23257 = 0.2907. The mean is 10 × 1/6 = 1.67 sixes and the standard deviation is 1.18.
The four conditions are a fixed number of trials, two outcomes per trial, the same p on every trial, and independent trials. Drawing cards without putting them back breaks the third and fourth conditions, so the binomial formula does not apply.
Mean, Variance and Large n
The calculator works in logarithms, so it stays accurate for large numbers of trials where the coefficients overflow ordinary arithmetic. When n is large and np and n(1 − p) are both at least about 10, the distribution is close to a normal one with the same mean and standard deviation, and the Normal Distribution Calculator gives a quick approximation.
For the chance of at least one success in n tries, the Probability Calculator has a dedicated mode.
How to Use the Binomial Probability Calculator
- Enter the number of trials: Type n, the number of independent trials.
- Enter the probability of success: Type p as a decimal between 0 and 1.
- Enter the number of successes: Type k. All five probabilities for k appear at once.
- Add a range: Optional. Type an upper number to get the probability of between k and that number of successes.
Frequently Asked Questions
What is the binomial probability formula?
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) is the number of ways to choose k successes from n trials. For 6 heads in 10 coin flips, that is 210 × 0.5^10 = 0.2051.
How do I calculate at least k successes?
Add P(X = x) for every x from k to n, or take 1 − P(X ≤ k − 1). For at least 6 heads in 10 flips the answer is 386 ÷ 1,024 = 0.3770.
What is the difference between at most and less than?
At most k includes k: P(X ≤ k). Less than k does not: P(X < k) = P(X ≤ k − 1). For 10 flips, P(X ≤ 6) = 0.8281 and P(X < 6) = 0.6230.
When can I use the binomial distribution?
When there is a fixed number of trials, each trial has two outcomes, the probability of success is the same on every trial, and the trials are independent.
What are the mean and standard deviation of a binomial distribution?
The mean is np and the variance is np(1 − p). For 10 flips of a fair coin the mean is 5, the variance 2.5 and the standard deviation 1.5811.
Is this the same as binompdf and binomcdf?
Yes. P(X = k) is what a TI calculator calls binompdf(n, p, k), and P(X ≤ k) is binomcdf(n, p, k).
Formulas and worked examples checked in October 2026.