How to use the binomial probability calculator
- Enter the number of trials n, a whole number from 1 to 10,000.
- Enter the probability of success p on each trial, as a decimal between 0 and 1.
- Enter the number of successes k you are interested in.
- Read the exact probability P(X = k), the cumulative probabilities, the mean and the standard deviation, and the table of probabilities around k.
Formula
P(X = k) = C(n, k) x p^k x (1 - p)^(n - k)
P(X <= k) = sum of P(X = i) for i from 0 to k
P(X >= k) = 1 - P(X <= k - 1)
Mean = n x p
Variance = n x p x (1 - p)
The terms are computed from logarithms of factorials so large n does not overflow, and the shorter tail is summed so tiny probabilities stay accurate.
Worked example
Flip a fair coin 10 times. The chance of exactly 6 heads is C(10, 6) / 1,024 = 210 / 1,024 = 0.205078. The chance of 6 or fewer is 848 / 1,024 = 0.828125, and of 6 or more is 0.376953.
The mean is 5 and the standard deviation is 1.581139. With 100 trials at p = 0.1, the chance of exactly 10 successes is 0.131865.
When a binomial model fits
Use it when there are a fixed number of independent trials, each with two outcomes and the same chance of success, such as coin flips, defective items in a batch from a large process or yes or no survey answers. If you sample without replacement from a small group, the chance changes with each draw and a hypergeometric model is more accurate.
Exact and cumulative probabilities
P(X = k) is the chance of exactly k successes. The cumulative values add up outcomes: at most k, fewer than k, at least k and more than k. "At least" questions are common, such as the chance of 8 or more successful sales calls out of 10. For counting arrangements by hand, use the combinations and permutations calculator.
Related statistics
For large n, the binomial is close to a normal curve, which you can study with the z-score calculator. To describe the spread of a set of results, use the standard deviation calculator, and for an interval estimate of a proportion see the confidence interval calculator.
Assumptions and limits
- There are n independent trials with the same probability p of success.
- n is a whole number up to 10,000 and k is a whole number from 0 to n.
- Probabilities smaller than 0.000001 are shown in scientific notation.
- Dependent trials, changing probabilities and sampling without replacement are not modeled.
Frequently asked questions
How do I calculate binomial probability?
Multiply the number of ways to pick k successes from n trials, C(n, k), by p to the k and by (1 - p) to the n minus k. For 6 heads in 10 fair flips that is 210 times 0.5 to the tenth.
What is the difference between P(X = k) and P(X <= k)?
P(X = k) is the chance of exactly k successes. P(X <= k) is the chance of k or fewer, which adds the probabilities of every count from 0 up to k.
What is the mean of a binomial distribution?
It is n times p. In 100 trials with p = 0.1, you expect about 10 successes on average.
Why does the calculator limit n to 10,000?
Beyond that the exact sums take longer and a normal approximation is usually good enough. The limit keeps the page fast on every device.