Two questions, two rules
Combining probabilities almost always comes down to one of two questions: do all of these happen, or does at least one of them happen. Each has its own rule, and they are easy to mix up.
The all-happen case multiplies the probabilities directly. The at-least-one case is easier to reach through the back door: find the chance that nothing happens, then subtract from one.
A worked example
Three independent events with probabilities 0.1, 0.2 and 0.3:
| All three happen | 0.1 × 0.2 × 0.3 | 0.006 |
|---|---|---|
| None happen | 0.9 × 0.8 × 0.7 | 0.504 |
| At least one happens | 1 − 0.504 | 0.496 |
Notice that adding 0.1 + 0.2 + 0.3 gives 0.6, which is wrong: it double-counts the cases where more than one happens. The complement method avoids that trap.
Why rare things happen eventually
The at-least-one rule is the mathematics behind a lot of everyday risk. A one-in-a-hundred event is unlikely on any single day, but over a year of independent days the chance of it happening at least once is 1 − 0.99^365 ≈ 97%. Repetition is what turns unlikely into near-certain.
Frequently asked questions
How do I find the probability that all events happen?
For independent events you multiply their probabilities. Two coin flips both landing heads is 0.5 × 0.5 = 0.25. Three events at 0.1, 0.2 and 0.3 all happening is 0.1 × 0.2 × 0.3 = 0.006, a little over half a percent.
How do I find the probability of at least one?
Work out the chance that none happen and subtract from one. For 0.1, 0.2 and 0.3 the chance each does not happen is 0.9, 0.8 and 0.7, so none happening is 0.9 × 0.8 × 0.7 = 0.504, and at least one is 1 − 0.504 = 0.496. Adding the individual probabilities instead would double-count the overlaps and overstate the answer.
What does independent mean here?
It means one event's outcome does not affect another's. Coin flips and dice rolls are independent. Drawing cards without replacement is not, because each draw changes what is left. This calculator assumes independence; for dependent events the multiplication no longer holds.
Can I mix percentages and decimals?
Enter each value in whichever form is clearer — 25% or 0.25 mean the same thing to the calculator. Every probability has to sit between 0 and 1 (0% and 100%); anything outside that is not a probability and will be flagged.
Why is 'at least one' so much higher than each single chance?
Because you get many chances for it to happen. Something with a 10% chance per try is far from certain once, but over ten independent tries the chance of at least one success is 1 − 0.9^10 ≈ 65%. This is exactly why rare risks become likely when repeated enough times.