| Binomial coefficient C(n, k) | 120 |
| P(X = 3) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ | 0.11718750 |
| P(X ≤ 3) | 0.17187500 |
| P(X ≥ 3) | 0.94531250 |
| P(X < 3) | 0.05468750 |
| Mean, μ = np | 5.000000 |
| Variance, σ² = np(1 − p) | 2.500000 |
| Standard deviation, σ | 1.581139 |
Each trial is independent and has the same success probability p. The number of ways to pick which k of the n trials succeed is C(n, k) = n! / (k! (n − k)!). Multiply that count by the probability of one specific pattern, pᵏ(1 − p)ⁿ⁻ᵏ, to get P(X = k). Add the terms to get cumulative probabilities: P(X ≤ k) sums from 0 to k, and P(X ≥ k) sums from k to n. For example, with n = 10 and p = 0.5, P(X ≤ 3) = (1 + 10 + 45 + 120) / 1024 = 176 / 1024 = 0.171875.
Definition and mean, variance: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.18.
Exact and cumulative binomial probabilities. The binomial distribution gives the probability of exactly k successes in n independent trials when each trial succeeds with the same probability p. This calculator returns the exact P(X = k), the cumulative probabilities P(X ≤ k) and P(X ≥ k), the mean np and the variance np(1 − p), with the binomial coefficient shown so each term can be checked.
Binomial probabilities assume the trials are independent and p is the same every time. Drawing without replacement from a small finite population changes p as you go, and then the hypergeometric distribution is the right model. 100% free, no registration, and complete privacy — everything runs locally in your browser, so your data never touches a server.
C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ, with the binomial coefficient computed exactly as an integer.
P(X ≤ k) and P(X ≥ k) are summed term by term, so the far tail keeps its precision.
μ = np and σ² = np(1 − p), with the standard deviation, for a quick check on the spread.
With n = 10 and p = 0.5, P(X ≤ 3) = 176/1024 = 0.171875, which matches the exact count of outcomes.
Exact nPr, nCr, factorials and counts with repetition.
Event probability, complements, P(A or B) and conditional probability.
Standard normal probabilities for z from 0 to 3.49.
Flip a fair coin online, one flip or up to 1,000, with heads and tails counts.
P(X = k) is the chance of exactly k successes. P(X ≥ k) adds the chances of k, k + 1, up to n successes. For a test of whether the result is better than chance, the cumulative tail is usually the quantity you need.
Use it when there is a fixed number of trials, each trial has two outcomes, the probability of success is the same every time, and the trials are independent. Coin flips and independent pass or fail tests fit. Draws without replacement from a small deck usually do not.
Exact probabilities for many trials involve very large coefficients, and the page sums each term in the browser. A limit of 1,000 keeps the calculation fast and accurate in double precision for practical inputs.
Yes, as a comparison. For large n, when np and n(1 − p) are both at least about 10, the normal distribution with mean np and variance np(1 − p) approximates the binomial well, with a continuity correction for tail probabilities.
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