Math
Probability Calculator
Enter the probability of each event to see how they combine.
The formula
both = A × B
either = A + B − (A × B)
neither = (1 − A) × (1 − B)
exactly one = A + B − 2 × (A × B)
Worked example
A 50% chance of rain and an independent 30% chance of a delayed train
- 0.50 × 0.30 = 15% both
- 0.50 + 0.30 − 0.15 = 65% at least one
- 0.50 × 0.70 = 35% neither
Where this goes wrong
Adding probabilities that can both happen
A 50% chance and a 30% chance do not give an 80% chance of at least one. Adding counts the overlap twice, so you have to subtract the 15% where both occur, leaving 65%. The giveaway is when the sum passes 100% — two 60% chances cannot make a 120% chance of anything.
Questions
- What does independent mean here?
- That one event happening tells you nothing about the other. Rain and a train delay are usually not independent — rain causes delays — and for events like those these formulas understate the joint probability.
- Why is "neither" not just 100% minus "both"?
- Because between them sits the case where exactly one happens. The four outcomes — both, only A, only B, neither — are what add to 100%, not just the two extremes.
- Can I chain more than two events?
- For all of them happening, yes: keep multiplying. For at least one, it is easier to work out the probability that none happen and subtract from one, which avoids counting overlaps at all.