At Least One Probability Calculator
Find the probability that an event happens at least once in n independent trials using the complement rule P = 1 − (1 − p)ⁿ.
How it works
P(at least one) = 1 − (1 − p)ⁿ
The direct way — adding the chances of exactly one, exactly two, and so on — is painful, so we use the complement. The only way to get zero successes is to fail every trial, which has probability (1 − p)ⁿ for independent trials. Subtracting that from one gives the probability of one or more successes, and it climbs quickly as n grows even when p is small.
Worked example
For a 20% chance per trial over 5 trials: P(none) = 0.8⁵ = 0.32768, so P(at least one) = 1 − 0.32768 = 0.67232, about 67%.
Sources: Complementary event — Wikipedia
Frequently asked questions
Why use the complement instead of adding directly?
There is only one way to get zero successes but many ways to get one or more. Computing the single “all failures” case and subtracting is far simpler and gives the exact answer.
Do the trials have to be independent?
Yes, this formula assumes each trial is independent with the same probability p. If trials influence each other, you need a tree or conditional model instead.
How many trials to be 99% sure of at least one success?
Solve 1 − (1 − p)ⁿ ≥ 0.99, i.e. n ≥ ln(0.01) / ln(1 − p). For p = 20% that is about 21 trials.
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This calculator is for educational purposes. Double-check important results before acting on them.