Probability
Binomial Distribution Calculator FAQ
Exact probabilities, tails, range checks, a distribution chart, CSV export, and binomial-test p-values in one worksheet.
FAQ
What does P(X = k) mean?+
It is the exact probability of getting exactly k successes in n independent trials when each trial has a constant success probability p. For example, the probability of getting exactly 6 heads in 10 fair coin flips would be P(X = 6) with n=10 and p=0.5.
What is the binomial probability formula?+
The formula is P(X = k) = C(n,k) × p^k × (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the success probability per trial, and C(n,k) is the binomial coefficient (n choose k) counting how many ways k successes can occur in n trials.
What is the difference between P(X <= k) and P(X >= k)?+
P(X ≤ k) is the lower-tail cumulative probability of k or fewer successes, which sums P(X=0) through P(X=k). P(X ≥ k) is the upper-tail cumulative probability of k or more successes, which sums P(X=k) through P(X=n). These are used for one-sided hypothesis tests and confidence intervals.
Can I calculate a range like P(3 <= X <= 7)?+
Yes. Enable range mode and enter the lower and upper success counts. The calculator sums each exact binomial probability for every integer in that inclusive range, giving you the total probability that the number of successes falls between your specified bounds.
When can I use the normal approximation?+
The normal approximation to the binomial is generally reliable when np ≥ 5 and n(1-p) ≥ 5. The calculator shows you whether this condition is met, lets you toggle the continuity correction (adding or subtracting 0.5 before calculating the z-score), and displays the approximation alongside the exact value so you can compare them directly.
Is the A/B test mode a binomial test?+
Yes. The A/B test mode treats the observed number of successes as a binomial count under a hypothesized success rate p0. It computes exact one-sided (greater-than or less-than) and two-sided p-values, which tell you how likely the observed result would be if the true rate were actually p0.
What is the continuity correction in the normal approximation?+
The continuity correction adjusts the discrete binomial distribution to better match the continuous normal distribution by adding or subtracting 0.5 from k before computing the z-score. The formula becomes z = (k ± 0.5 - np) / sqrt(np(1-p)), which typically improves the approximation accuracy, especially for small n or extreme p.