Probability Calculator
Combine two independent event probabilities: and, or, neither.
Assumes A and B are independent. For events that affect each other, these formulas do not apply directly.
Combining independent chances
When two events are independent, their combined probabilities follow a few clean rules. Multiplying gives the chance of both; a little care with overlap gives the chance of either.
| Both | P(A) × P(B) |
|---|---|
| At least one | P(A) + P(B) − P(A)×P(B) |
| Neither | (1 − P(A)) × (1 − P(B)) |
The key word is independent: each event must leave the other’s odds unchanged. Drawing cards without replacing them, for instance, breaks that assumption and needs conditional probability instead.
With two independent 50% events, both happening is 25% (0.5 × 0.5), at least one is 75%, neither is 25%, and exactly one is 50%.
Intuition that helps
Probabilities of "both" shrink fast — two coin flips landing heads is already only one in four. "At least one", by contrast, climbs quickly as you add events, which is why rare mishaps become likely over many tries.
Quick checks
- Both ≤ either input. Combining can only lower the joint chance.
- At least one ≥ either input. More ways to succeed.
- Neither + at least one = 100%. A handy sanity test.