The short answer
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain): P(A) = favorable outcomes ÷ total outcomes. For combining two events, use the right rule for how they relate: multiply for "and" with independent events, add and subtract overlap for "or," and divide for "given that" (conditional probability).
Key takeaways
- P(A and B) = P(A) × P(B) only works when A and B are independent — dependent events need conditional probability instead.
- P(A or B) = P(A) + P(B) − P(A and B) — the subtraction avoids double-counting outcomes that satisfy both events.
- Conditional probability, P(A|B) = P(A and B) ÷ P(B), narrows the "total outcomes" down to just the cases where B already happened.
- The complement rule — P(at least one) = 1 − P(none) — is often the fastest way to solve "at least one" problems that would otherwise need many separate cases added together.
The core probability rules
| Rule | Formula | Used for |
|---|---|---|
| Basic | P(A) = favorable/total | A single event |
| Union | P(A or B) = P(A)+P(B)−P(A∩B) | Either event happens |
| Joint (independent) | P(A and B) = P(A) × P(B) | Both happen, unrelated events |
| Conditional | P(A|B) = P(A∩B)/P(B) | One event given another already happened |
Worked example: rolling at least one six
What's the probability of rolling at least one six in 4 rolls of a fair die?
P(no six on one roll) = 5/6
P(no six in 4 rolls) = (5/6)⁴ ≈ 0.482
P(at least one six) = 1 − 0.482 ≈ 0.518 (51.8%)
Trying to solve this by directly adding up "exactly one six," "exactly two sixes," and so on would take far more work — the complement rule turns a multi-case problem into a single subtraction.
Independent vs. dependent events
Two coin flips are independent — the first flip's result has zero effect on the second. But drawing two cards from a deck without replacement is dependent — if the first card is a heart, there's one fewer heart left for the second draw, which changes its probability. Whenever removing or using up part of the sample space changes the odds for what comes next, the events are dependent, and conditional probability (not simple multiplication) is the right tool.
Common mistakes to avoid
- Adding P(A) + P(B) for "or" without subtracting the overlap P(A and B) — this double-counts outcomes that satisfy both events.
- Multiplying P(A) × P(B) for events that aren't actually independent — that shortcut only applies when one event has no effect on the other.
- Confusing P(A|B) with P(B|A) — these are generally different values, a mix-up behind many real-world misreadings of medical test results and statistics.
- Ending up with a probability outside the 0-to-1 range — that always signals an error upstream, since a valid probability can never be negative or greater than 1.
Related calculators
- Permutation & Combination Calculator — count the favorable and total outcomes that feed into a probability.
- Statistics Calculator — analyze real data behind an experimental probability estimate.
- Sample Size Calculator — determine how many trials you need to estimate a probability reliably.
- Confidence Interval Calculator — express an estimated probability as a range instead of a single number.