The three combining rules
Once you know the chance of each event, basic probability rules let you combine them. The chance that both A and B happen is the product of their individual chances — but only when the events are independent. The chance that at least one happens adds the two probabilities and then subtracts the overlap, so the shared outcome is not counted twice. The chance that A does not happen is simply one minus the chance that it does.
The chart lines these results up next to the inputs so you can see how "and" is always the smallest, "or" the largest, and the complement mirrors P(A).
Why "or" subtracts the overlap
If you just added P(A) and P(B), any outcome where both occur would be tallied in each term, double-counting it. The product P(A)·P(B) is exactly that overlap for independent events, so subtracting it once gives the correct "at least one" probability.
- P(A and B) can never exceed either input on its own.
- P(A or B) can never exceed 1, and equals P(A) + P(B) only when the events cannot both happen.
- P(not A) and P(A) always sum to exactly 1.
When the independence assumption breaks
These formulas assume the two events do not influence each other. That holds for separate coin flips or dice rolls, but not for events that are linked — drawing two cards without replacement, or weather on consecutive days. When events are dependent you need conditional probability, where P(A and B) = P(A)·P(B given A).
Before trusting the "and" result, ask whether knowing the outcome of one event would change your estimate of the other. If it would, the multiplication rule used here does not apply.
Formula
P(A and B) = P(A)·P(B); P(A or B) = P(A) + P(B) − P(A)·P(B); P(not A) = 1 − P(A)Frequently asked questions
- What does "independent events" mean?
- Two events are independent when the outcome of one does not affect the other. The multiplication rule P(A and B) = P(A)·P(B) only holds in that case.