At Least One Probability Calculator
The probability of getting at least one success in n independent trials is one of the most fundamental results in probability theory. This calculator computes it using the complement rule: P(at least one) = 1 minus P(none), where P(none) = (1 minus p)^n. Enter the per-trial success probability and number of trials to get the cumulative probability. The calculator also shows how many trials you need for 50%, 90%, and 99% cumulative probability at your given per-trial rate.
At least one probability formula
P(at least one) = 1 - (1 - p)^n
Trials for target T: n = ln(1 - T) / ln(1 - p)
The complement rule converts P(zero successes) = (1 minus p)^n into the at-least-one probability. All n trials are assumed to be independent with the same probability p.
Applications in probability and game design
- A 10% per-trial rate gives 65.1% cumulative probability after 10 trials, not 100% as simple multiplication would suggest.
- For a 1% rare item with 460 attempts, the cumulative probability exceeds 99%, meaning almost everyone will see the item by that point.
- Anti-virus software vendors use the same formula to estimate the probability of at least one malware variant evading detection across thousands of scan attempts.
- Game designers use this formula to decide how many boss kills are needed before the vast majority of players obtain a rare item.
At least one probability: frequently asked questions
What is the at-least-one probability formula?
P(at least one success) = 1 - (1 - p)^n, where p is the probability of success on a single trial and n is the number of independent trials. This is the complement of the probability of failing every single trial.
When should I use this formula?
Use it any time you want to know the cumulative probability of an event occurring at least once over repeated independent attempts. Examples include loot drop rates, critical hit chances over multiple attacks, quest completion chances, and gacha pulls without pity.
What is the difference between this and just multiplying p by n?
Multiplying p by n gives an approximation that only works for very small p and n. The correct formula is 1 minus (1 minus p)^n, which accounts for the decreasing probability of all failures as n increases. For p=0.5 and n=2, p*n would give 1.0, which is wrong; the correct answer is 0.75.
How many trials do I need for a 90% cumulative probability?
Solve n = ln(1 - 0.9) / ln(1 - p) = ln(0.1) / ln(1 - p). For p=0.1 (10% per trial), you need about 22 trials for 90% cumulative probability. The calculator shows this automatically.
Can the probability exceed 100%?
No. The formula always returns a value between 0 and 1 (0% to 100%), approaching but never exceeding 100% for finite n. The probability becomes essentially 100% for large n relative to 1/p.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: itl.nist.gov/div898/handbook/.
- National Council of Teachers of Mathematics: nctm.org.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.