Binomial Probability Calculator
Enter the number of trials n, the number of successes k and the chance of success p to get P(X = k), P(X at most k), P(X at least k), the strict versions, the mean, the standard deviation and the most likely value. A bar chart shows the chance of every number of successes. You can type p as a decimal, a percent or a fraction. Everything runs in your browser.
Chance of each number of successes
- 00.000976563
- 10.00976563
- 20.0439453
- 30.117188
- 40.205078
- 50.246094
- 60.205078
- 70.117188
- 80.0439453
- 90.00976563
- 100.000976563
The binomial distribution counts the successes in n independent trials that each succeed with the same probability p, such as coin flips or free throws. The chance of exactly k successes is n choose k, times p to the power of k, times (1 - p) to the power of (n - k). The cumulative values add up the exact chances, tail by tail, so even a very small probability keeps its digits. The trials must be independent and p must stay the same, which is not true of drawing cards without putting them back. Everything is worked out in your browser and nothing is uploaded.
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Writing & ContentAbout the Binomial Probability Calculator
The binomial distribution counts the successes in n independent trials that each succeed with the same probability p, such as coin flips, free throws or defective parts in a sample. The chance of exactly k successes is n choose k times p to the power of k times (1 - p) to the power of (n - k). Ten flips of a fair coin give 5 heads with probability 0.24609375, which is about 24.6 percent. The cumulative values add up the exact terms: at most k adds the chances of 0 to k, and at least k adds the chances of k to n. Each tail is summed directly instead of subtracted from 1, so a very small probability keeps its significant digits. The terms are computed in log space, so n up to 10,000 does not overflow. The mean is n times p, the variance is n times p times (1 - p), and the most likely value is the whole number part of (n + 1) times p. The model needs independent trials and a constant p, so it does not describe drawing cards without putting them back. The results were checked against sums of exact binomial coefficients and against a fair coin, where 100 flips give 60 or more heads with probability 0.02844. Nothing is uploaded.
How do I calculate a binomial probability?▾
Use n choose k times p to the power of k times (1 - p) to the power of (n - k). For 5 heads in 10 fair coin flips that is 252 times 0.5 to the power of 10, which is 0.2461. Enter n, k and p and the tool does the sum.
What is the difference between exactly, at most and at least?▾
Exactly k is one term. At most k adds the chances of 0 through k. At least k adds the chances of k through n. The strict versions, fewer than and more than, leave out k itself.
When can I use the binomial distribution?▾
When there is a fixed number of trials, each trial has two outcomes, the chance of success stays the same, and the trials are independent. Coin flips and dice rolls qualify. Drawing cards without replacement does not, because the chances change.
What are the mean and standard deviation?▾
The mean is n times p, the number of successes you expect on average. The standard deviation is the square root of n times p times (1 - p). For 100 flips of a fair coin the mean is 50 and the standard deviation is 5.
How large can n be?▾
Up to 10,000 trials. The terms are computed with logarithms, so large values do not overflow, and the chart shows the 41 values around the mean when n is large. Nothing you enter is uploaded.