Probability of At Least One Calculator
Chance that an event happens at least once over repeated independent tries, with the tries needed for even odds.
Results
What this tool does
Small risks add up in a way that feels wrong: something with a one percent chance per attempt is more likely than not to have happened after sixty-nine attempts. The last two results give the number of tries for a fifty percent and a ninety-five percent chance, which is usually the question behind the question.
Formula
P = 1 − (1 − p)^n
Variables
| Symbol | Meaning | Unit |
|---|---|---|
pp | Probability each time | % |
nn | Number of tries | — |
PA | Probability | % |
PN | Probability | % |
EX | True or expected value | — |
N5 | Number of tries | — |
N9 | Number of tries | — |
Worked example
- Probability each time5 %
- Number of tries20
- Probability64.15141 %
- Probability35.84859 %
- True or expected value1.0000
- Number of tries13.51
- Number of tries58.40
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
Frequently asked questions
Why is a one-in-twenty risk over twenty tries not certain?
Because each try is independent, and the chances of failing every time never quite vanish. Twenty tries at five percent gives 64 percent, not 100 — and the number it converges towards as the tries grow is about 63.2 percent, which is one minus one over e. The expected count of successes is exactly one, but expecting one on average is not the same as getting one.