Binomial probability calculator
The binomial distribution counts successes in a fixed number of independent trials, each with the same probability.
%
Probability of exactly k successes
11.718 750%
- Probability of at least one success
- 99.902 344 %
- Probability of no successes
- 0.097 656 %
- Expected number of successes
- 5.0000
- Standard deviation
- 1.581 14
- Variance
- 2.500 00
- Z-score of k
- 1.2649
Show the working
- Number of trials = 10
- Number of successes = 7
- Probability of success = 50
- prob = 50 / 100 = 0.5
- Probability of exactly k successes = 7 <= 10 ? comb(10, 7) * 0.5^7 * (1 - 0.5)^(10 - 7) * 100 : 0 = 11.7188
- Expected number of successes = 10 * 0.5 = 5
- Variance = 10 * 0.5 * (1 - 0.5) = 2.5
- Standard deviation = sqrt(2.5) = 1.581 14
- Z-score of k = 1.581 14 > 0 ? (7 - 5)/1.581 14 : 0 = 1.264 91
- Probability of no successes = (1 - 0.5)^10 * 100 = 0.097 656 3
- Probability of at least one success = 100 - 0.097 656 3 = 99.9023
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- P(X = k) = C(n,k) p^k (1-p)^(n-k)
- mean = np, variance = np(1-p)
The binomial coefficient is computed as an exact integer, so this stays correct for large n where a float would overflow.