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Binomial Probability Calculator

Binomial probabilities for k of n trials

Result

26.68%
61.72%

How it is calculated

For n independent trials, each with the same success probability p, the chance of exactly k successes is P(X = k) = C(n, k) × pk × (1 − p)n−k, where C(n, k) = n! ÷ (k! × (n − k)!). The second result adds this up from k to n: P(X ≥ k). The probability p is entered as a percentage from 0 to 100, not as a fraction, and the tool divides it by 100 itself. Needs n ≥ 1 and 0 ≤ k ≤ n. Decimals in n and k are cut to whole numbers, so 3.7 is treated as 3. The tool does not give P(X ≤ k) directly; for it use 100% minus the at-least value for k + 1.

Worked example

Ten trials, a 30% chance each, and we look at 3 successes. The number of ways to place 3 successes among 10 trials is C(10, 3) = 120. Then p3 = 0.33 = 0.027 and (1 − p)7 = 0.77 ≈ 0.0824. Multiply: 120 × 0.027 × 0.0824 ≈ 0.2668, so exactly 3 successes has probability 26.68%. Adding the cases 3 through 10 gives the chance of at least 3 successes: 61.72%.

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Common mistakes

  • Entering p as a fraction. Typing 0.3 means 0.3%, and the result is almost 0. For 30%, type 30.
  • Reading "at least k" as "exactly k" or the other way round. The first includes every larger outcome.
  • Using the tool when trials are not independent, such as drawing cards without replacement from a small deck. The formula needs a fixed p on every trial.
  • Mixing up n (trials) with k (successes) so that k is larger than n, which gives no result.
  • Expecting the most likely outcome to have a high probability. Even the single most likely count is often below 30%, as the exact result shows.

About this calculator

For n independent trials each with success probability p, this calculator computes P(X=k) = C(n,k) × p^k × (1−p)^(n−k) — the binomial probability mass function — for an exact number of successes, and sums it across values to get P(X≥k) or P(X≤k) for at-least or at-most probabilities.

It's the standard model whenever an outcome is binary and trials are independent with a fixed success rate: quality control engineers use it to find the probability of k defective units in a batch, geneticists apply it to Mendelian inheritance ratios, pollsters and A/B testers use it to judge whether an observed split could plausibly arise by chance, and it's a core topic in every introductory statistics course.

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