Three Event Probability Calculator
The chance that at least one of three independent events happens, that all three happen, and that exactly one or exactly two do.
For independent events the "all three" case is the product P(A)P(B)P(C), and the "none" case is the product of the complements.
How the three event probability calculator works
For independent events the "all three" case is the product P(A)P(B)P(C), and the "none" case is the product of the complements. Everything else follows: at least one is 1 − none, and the exactly-k counts come from summing the ways to pick which succeed.
The practical lesson is how quickly small risks compound: three independent 10% risks give a 27.1% chance that at least one occurs.
Formula: P(at least one) = 1 − (1 − a)(1 − b)(1 − c)
Worked examples
| Inputs | P(at least one) | Note |
|---|---|---|
| Three 10% risks | 27.1% | 27.1% that at least one happens |
| Three coin flips | 87.5% | 87.5% at least one head |
| Three 1% risks | 2.9701% | 2.97% |
FAQFrequently asked questions
How do I combine three probabilities?
For independent events, multiply the complements and subtract from 1. Three 10% risks give 1 − 0.9³ = 27.1%.
Why not just add them?
Because that double- and triple-counts the overlaps, and it can exceed 100%. Three 50% events would "add" to 150%.
What does independent mean here?
That none of the events changes the odds of the others. If they share a cause, this calculation understates the joint outcomes.
What is the expected number?
Simply the sum of the three probabilities — that part is true whether or not they are independent.
Why do small risks feel bigger in combination?
Because compounding is multiplicative in the complements. Ten independent 1% risks give a 9.6% chance that at least one occurs.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.