Probability of At Least One Event in Repeated Trials: Rain, Defects, and Cumulative Risk
October 29, 2026
A small probability repeated many times adds up. If an event has probability p on each of n independent trials, the probability that it happens at least once is 1 − (1 − p)^n. The trick is to compute the probability that it never happens, (1 − p)^n, and subtract from 1. With a 30% chance of rain each day over a 7-day week, the chance of no rain at all is 0.7⁷ = 8.2%, so the chance of at least one rainy day is 91.8%.
The result is often surprising. A 5% risk per year, repeated for 20 years, creates a 64.2% chance of at least one occurrence. A 1-in-100 chance every day for 100 days gives 63.4%.
Skip the math and get your answer instantly:
Open the Percentage Calculator →Table of Cumulative Probabilities
The table gives 1 − (1 − p)^n for several cases. Notice that a 1-in-n chance repeated n times gives about 63.2% for large n (the limit is 1 − 1/e = 0.632), not 100%. To reach 95% certainty of at least one event, you need n ≈ 3 ÷ p trials (the 'rule of three'): for p = 1%, about 300 trials.
For small p, the approximation 1 − (1 − p)^n ≈ n × p works when n × p is small: ten trials at 1% gives about 9.6% (exactly 9.56%), close to 10%.
| Scenario | p per trial | Trials n | At least one |
|---|---|---|---|
| Rain, 30% daily | 30% | 7 days | 91.8% |
| Six rolls of a die, at least one 6 | 16.7% | 6 | 66.5% |
| Risk of 10% per year | 10% | 10 years | 65.1% |
| Risk of 5% per year | 5% | 20 years | 64.2% |
| 1% chance per day | 1% | 100 days | 63.4% |
| 1% chance per day | 1% | 300 days | 95.1% |
Examples From Daily Life
Defect inspection: if each item has a 2% defect rate, a batch of 50 has a 1 − 0.98⁵⁰ = 63.6% chance of containing at least one defective. Security: a password breach risk of 2% per year over 10 years gives 18.3%. Medical screening: a test with a 5% false-positive rate repeated 10 times on a healthy person produces at least one false positive with probability 40.1%. Insurance: if your chance of a car accident is 3% a year, the chance of at least one in 30 years of driving is 59.9%. Flooding: a '100-year flood' has a 1% chance each year, and a 26% chance of occurring at least once in 30 years, which is the length of a mortgage.
When Independence Fails
The formula assumes the trials are independent and p stays constant. Rain on consecutive days is correlated, so the real chance of at least one rainy day in a week may be lower than 91.8% in a dry climate (when dry spells persist) or higher in wet seasons. Risks like illness or financial shocks can be positively correlated, and conditions that change over time, like aging, alter p. With dependence, cumulative probabilities need other models, but the formula is still a useful first approximation.
Using It for Decisions
To keep the cumulative risk below a target q over n trials, require p ≤ 1 − (1 − q)^(1/n). To limit the lifetime risk of at least one failure to 5% over 100 uses, each use must have a failure chance below 1 − 0.95^(0.01) = 0.051%. This is why safety-critical systems need very high reliability per operation, and why small risks deserve attention when they are repeated often.
Frequently Asked Questions
How do I calculate the probability of at least one success in n trials?
1 − (1 − p)^n.
What is the chance of at least one 6 in six rolls?
About 66.5%.
Does a 1-in-100 chance repeated 100 times guarantee an event?
No. It gives about a 63% chance.
What is the rule of three?
About 3/p trials are needed for roughly 95% chance of seeing an event with probability p.
What does independence mean here?
That one trial's outcome does not affect the others.