Binomial Distribution Calculator

When you need at least k successes

Most practical decisions hinge on a threshold, not one exact count: enough working units, enough completed calls, enough caught defects. The Binomial Distribution Calculator answers that cumulative question, giving the probability of at least your chosen number of successes across n independent trials with success rate p. QA leads, risk analysts, and campaign managers use it to quote odds they can defend.

Summing the binomial terms

An at-least-x probability adds P(X = k) for every k from x to n, each term being C(n, k) p^k (1-p)^(n-k). Example with n = 5 and p = 0.8: P(4) = 0.4096 and P(5) = 0.32768, so P(at least 4) = 0.73728. The same method gives P(at least 3) = 0.94208, a far more comfortable number for planning.

Steps for a cumulative question

Enter Probability (p) as a decimal, set Number of trials (n), and place the threshold in Number of events (x). Press Calculate and the Desired probability sums every count from your threshold up to n. Watch the wording of your question carefully: at least and exactly differ by roughly 0.33 in this example, a gap big enough to change decisions.

A fresh scenario shows how the threshold changes the odds with a different n. Take a quiz with Number of trials (n) = 10 true false questions, Probability (p) = 0.5 per question, and Number of events (x) = 8. The terms are P(8) = 0.0439, P(9) = 0.0098, and P(10) = 0.0010, so the Desired probability for at least 8 lands near 0.0547. Purely guessing your way to 8 correct answers out of 10 is a 5.5 percent long shot, which is why studying beats coin flips every time.