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Binomial Distribution Calculator

Find the probability of k successes in n trials.

How it is calculated

The binomial distribution counts successes in n independent trials that each succeed with probability p. The exact probability of k successes is P(X=k) = C(n,k) × pk × (1−p)n−k, where C(n,k) = n! / (k!(n−k)!). The tool also returns the cumulative P(X ≤ k), the sum of the exact probabilities from 0 to k, and P(X ≥ k), which equals 1 − P(X ≤ k) + P(X=k). Here p is a probability between 0 and 1 (not a percentage), n is a whole number up to 1000 and k must lie between 0 and n. Probabilities are shown to five decimals.

Worked example

Flip a fair coin 10 times (n = 10, p = 0.5) and ask for exactly 3 heads (k = 3). C(10,3) = 120, and 0.5³ × 0.5⁷ = 1/1024, so P(X=3) = 120/1024 = 0.11719. Cumulative: the exact chances for 0, 1, 2 and 3 heads are 1, 10, 45 and 120 out of 1024, totalling 176/1024 = 0.17188 for at most 3 heads. At least 3 heads is 1 − 0.17188 + 0.11719 = 0.94531.

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Common mistakes

  • Entering p as a percentage (50) instead of a probability (0.5).
  • Confusing "exactly k" with "at most k" or "at least k"; they are three separate outputs.
  • Using the model when trials are not independent or the success chance changes, such as drawing cards without replacement.
  • Exceeding n = 1000 or setting k above n, which gives no result.
  • Reading "at least k" as 1 − P(X ≤ k); that excludes k itself, so it understates the answer.

About this calculator

The binomial distribution models the number of successes in n independent trials, each with the same success probability p. The exact probability of getting precisely k successes is P(X=k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ, where C(n,k) is the number of ways to choose which k trials succeed; this calculator also sums that formula to give the cumulative probabilities P(X≤k) and P(X≥k).

Statisticians, quality-control engineers, and data scientists use the binomial distribution whenever an outcome is binary and trials are independent — defect rates in manufacturing batches, coin-flip and dice problems in intro statistics, and conversion-rate comparisons in A/B testing all reduce to this model. Clinical researchers use it too, for example to compute the probability of a given number of patients responding to treatment out of a fixed trial size.

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Comments (1)

  • Editorial team

    A practical use is spot checks in quality control. Suppose each of 20 items has a 5% chance of being defective, so n = 20 and p = 0.05. Enter k = 2 and read the at-least line: the chance of finding two or more defective items is about 0.264. Use it only if items are independent. What kind of sampling or yes/no situation are you trying to model?

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