Probability Calculator

Worked example

A 20% event and a 30% event, independent: both at 6%, either at 44% — not 50%, because adding them double-counts the overlap. And a 20% event tried ten times happens at least once 89.3% of the time.

How to work it out yourself

1. 1.Enter the two probabilities as percentages. They do not have to add to 100 — these are separate events, not alternatives.
2. 2.Check whether the events are actually independent before trusting the "both" line. Rain today and rain tomorrow are not.
3. 3.Use the "at least once" line for repeated exposure: a small risk taken often is a large risk, and that line is where the intuition usually fails.

The formula

1. Both happen, A and B0.2 * 0.3 * 100
2. At least one happens, A or B(0.2 + 0.3 - 0.2 * 0.3) * 100
3. Neither happens(1 - 0.2) * (1 - 0.3) * 100
4. Exactly one happens(0.2 * (1 - 0.3) + 0.3 * (1 - 0.2)) * 100
5. A does not happen(1 - 0.2) * 100
6. A happens at least once in the tries(1 - pow(1 - 0.2, 10)) * 100
7. Expected number of times A happens0.2 * 10

Source: NIST/SEMATECH e-Handbook of Statistical Methods — probability

Questions people actually ask

Why is "A or B" not just A plus B?

Because adding them counts the overlap twice. 20% plus 30% is 50%, but the 6% where both happen has been included in each, so the answer is 44%. Subtracting the overlap is the inclusion-exclusion rule, and it is why probabilities added carelessly can exceed 100%.

What does independent mean?

That one event happening tells you nothing about the other. Two dice are independent; two flights on the same day in the same weather are not. Every line on this page assumes independence, and where it fails the "both" figure can be far wrong in either direction — correlated risks cluster.

How likely is a rare event over many tries?

Far likelier than people expect. A 1% chance repeated 100 times comes to 63.4%, not 100% and not 1%. The arithmetic is one minus the chance of it never happening, and it is the reason a small per-flight or per-year risk becomes a substantial lifetime one.

What is the complement rule?

The chance something does not happen is one minus the chance it does. It sounds trivial and it is the most useful trick in elementary probability: "at least one" questions are almost always easier solved as "one minus none", which is exactly how the line above is computed.

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