Binomial Cumulative Probability Calculator

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.

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.