Binomial Probability Calculator

How it is calculated

The tool gives the probability of exactly k successes: P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where n is the number of independent trials, k the number of successes and p the success probability of one trial, written as a decimal between 0 and 1. The coefficient C(n, k) = n! / (k! (n − k)!) counts the orders in which the successes can occur.

n and k are rounded down to whole numbers. The result is shown as a percentage, together with the expected number of successes n × p. No result is given if k is larger than n or p is outside 0 to 1. It is a single-value probability, not a cumulative one.

Common mistakes

• Entering p as a percentage (50) instead of a decimal (0.5); the tool then returns no result.
• Reading the answer as "3 or fewer" or "at least 3". It is the chance of exactly 3; for ranges, add the probabilities for each k.
• Using the model when trials are not independent or p changes, such as drawing cards without putting them back.
• Confusing the expected value (5 successes) with the most probable outcome for non-integer n × p; they can differ.