Probability of At Least One Event Calculator
Find the chance that an event happens at least once over several independent tries using the complement rule — 1 minus the probability it never happens.
How it works
P(at least one) = 1 − (1 − p)ⁿ · where p is the per-try probability and n the number of independent tries
It is far easier to compute the chance an event never happens than to add up every way it could happen once or more. If a single try fails with probability 1 − p, then n independent tries all fail with probability (1 − p)ⁿ, and the complement is the chance of at least one success. This complement trick is why rare per-try risks can still become likely over many repetitions.
Worked example
An event has a 10% chance each try. Over 12 tries the chance it never happens is (1 − 0.10)¹² = 0.9¹² ≈ 0.282, so the probability it happens at least once is 1 − 0.282 = 0.718 — about 71.8%.
Sources: Complementary event — Wikipedia
Frequently asked questions
Why not just multiply the probability by the number of tries?
Multiplying (0.10 × 12 = 1.2) can exceed 100%, which is impossible for a probability, and it double-counts outcomes where the event happens more than once. The complement method 1 − (1 − p)ⁿ handles the overlap correctly.
Do the tries have to be independent?
Yes — the formula assumes each try does not affect the others and every try has the same probability p. If tries are correlated or p changes, the result is only an approximation.
How many tries until the event is more likely than not?
Solve 1 − (1 − p)ⁿ ≥ 0.5, which gives n ≥ ln(0.5) ÷ ln(1 − p). For a 10% per-try chance that is about 7 tries.
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For education — check the assumptions (sample size, independence, distribution) before using these results in real decisions.