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Binomial Probability Calculator

Q: What is the difference between P(X < k) and P(X ≤ k)?

P(X ≤ k) includes exactly k successes; P(X < k) does not.

Find exact and cumulative binomial probabilities for any number of trials and successes, run an exact binomial test and get a Clopper-Pearson confidence interval for the observed proportion.

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Number of trials (n)Probability of success (p)Number of successes (k)

Results

P(X = 7)	0.0739
P(X < 7)	0.0577
P(X ≤ 7)	0.1316Cumulative probability
P(X > 7)	0.8684
P(X ≥ 7)	0.9423
Mean and SD	10.000 and 2.236np and √(np(1 − p))
Exact binomial test (two-sided)	0.263H₀: p = 0.5
Observed proportion	0.350 (95% CI 0.154 to 0.592)Clopper-Pearson exact interval

How to Use the Binomial Probability Calculator

1
Enter n, p and k
Trials, probability of success and number of successes.
2
Read the probabilities
Exactly k, at most k, at least k, and more.

The binomial distribution

The binomial distribution gives the probability of k successes in n independent trials, each with the same probability of success p, such as the number of patients responding to a treatment.

The exact binomial test compares an observed proportion with a hypothesised value without relying on the normal approximation, which makes it suitable for small samples.

How it is calculated

P(X = k) = C(n, k) pᵏ(1 − p)ⁿ⁻ᵏ; cumulative probabilities by summation
Two-sided exact test: sum of probabilities no larger than P(X = k)
Clopper-Pearson interval from beta quantiles; results match SciPy binomtest

Sources

NIST/SEMATECH e-Handbook: Binomial distribution