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At Least Once Probability Calculator

Chance of ≥1 success in n tries.

How it is calculated

Enter the chance of success on one trial as a percentage p (0 to 100) and the number of independent trials n as a whole number of at least 1. The tool works through the complement: the chance of no success in all n trials is (1 − p ÷ 100)n, so P = 1 − (1 − p ÷ 100)n. The answer is a percentage with 2 decimals. Trials must be independent with the same p every time. Empty boxes, p below 0 or above 100, and an n that is below 1 or not whole show no result. p = 0 gives 0.00 % and p = 100 gives 100.00 %. For large n the answer rounds to 100.00 % even though the true value is slightly below it.

Worked example

Each of 10 independent attempts has a 5 % chance of success, so p = 5 and n = 10. One attempt fails with probability 1 − 0.05 = 0.95. All 10 fail with probability 0.9510 ≈ 0.5987. At least one success therefore has probability 1 − 0.5987 ≈ 0.4013, which the tool prints as 40.13%. Compare that with the naive guess of 10 × 5 % = 50 %.

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Common mistakes

  • Multiplying p by n. Ten trials at 5 % give 40.13 %, not 50 %; the plain sum double-counts runs with several successes and passes 100 % for large n.
  • Entering 0.05 for 5 %. The box takes a percentage, so 0.05 means 0.05 % and the result is far too small.
  • Using it for dependent trials, such as drawing cards without putting them back. The formula assumes the same chance on every try.
  • Entering a fractional number of trials like 2.5. The count must be a whole number, otherwise no result is shown.
  • Entering the chance as a negative number or above 100. Those are impossible probabilities and are rejected.

About this calculator

When an event has probability p of happening on a single independent trial, the probability that it happens at least once across n trials is easiest to find through the complement: instead of adding up every way it could succeed once, twice, or more, you calculate the probability it never happens — (1−p)ⁿ — and subtract that from 1. So P(at least one) = 1 − (1−p)ⁿ.

This formula underlies risk assessments across many fields: reliability engineers use it to estimate the chance that at least one component fails over n operating cycles, epidemiologists apply it to find the odds that at least one person in a group of n tests positive, and security teams use it to model the probability that at least one intrusion attempt out of many succeeds against a system with a known per-attempt breach rate. It is also the reasoning behind gambling odds, like the chance of rolling at least one six in several dice rolls.

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