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Binomial Cumulative Probability Calculator

Enter trials n, probability p and count k for the binomial distribution.

How it is calculated

The tool adds the binomial probabilities for every count from 0 to k: P(X ≤ k) = Σ C(n, i) × pi × (1 − p)n−i, for i = 0 … k. n is the number of independent trials, p is the chance of success on one trial as a fraction between 0 and 1 (not a percentage), and k is the largest number of successes you accept. The terms are summed in log space, so a large n does not overflow; the tool accepts up to 1,000,000 trials. The second result is the single term P(X = k) and the third is the mean n × p. Inputs that cannot describe a real experiment show no result: a fractional n or k, n below 1, a negative k, k larger than n, or p outside 0 to 1. Probabilities are shown with four decimals.

Worked example

Take 20 independent trials with a 0.3 chance of success each, and ask for the probability of at most 8 successes. The tool adds the terms for i = 0 to 8 and the total is 0.8867. The single term for exactly 8 successes is C(20, 8) × 0.38 × 0.712 = 0.1144. The mean is 20 × 0.3 = 6.00. The chance of 9 or more successes is therefore 1 − 0.8867 = 0.1133.

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Common mistakes

  • Typing p as a percentage. A 30 % chance is entered as 0.3; 30 gives no result.
  • Reading the main result as the chance of exactly k. That is the separate pmf figure; the main result is the chance of k or fewer.
  • Needing at least k successes but not converting. Take 1 minus the cumulative probability at k − 1.
  • Applying it when trials are not independent or the chance changes between trials, such as drawing without replacement from a small batch.
  • Entering a k larger than n, or a decimal k or n, and expecting a rounded answer. These inputs are rejected instead of adjusted.

About this calculator

For a series of n independent trials that each succeed with probability p (like flipping a biased coin n times), the binomial cumulative probability P(X ≤ k) gives the chance of getting at most k successes in total. It's calculated by summing the individual binomial probabilities C(n, i) × p^i × (1−p)^(n−i) for every count i from 0 up to k, where C(n, i) is the number of ways to choose which i trials succeeded.

Cumulative binomial probabilities are the standard way to answer “at least” or “at most” questions in quality control (the probability that no more than k defective items appear in a batch), clinical trials and medical statistics (the chance that at most k patients respond to a treatment), and everyday probability problems like the odds of winning at least a certain number of games or coin flips out of a fixed number of attempts, where the distribution's mean is simply n × p.

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