The AND and OR Rules
Grade 5The AND rule — both events happen
When you want the probability of event A and event B both happening, and the two events are independent (one doesn't affect the other), you multiply:
P(A and B) = P(A) × P(B)
Independent means the result of A has no effect on the probability of B — like flipping a coin twice, or rolling two separate dice.
The OR rule — at least one event happens
When you want the probability of A or B happening, and the events are mutually exclusive (they can't both happen at the same time), you add:
P(A or B) = P(A) + P(B)
Mutually exclusive means if one happens, the other can't — like rolling a 3 or a 5 on a single dice.
Worked example
EXAMPLE
A fair dice is rolled and a fair coin is flipped. Find (a) P(6 and heads), (b) P(odd number or tails).
(a) These are independent events. P(6) = 1/6, P(heads) = 1/2. P(6 and heads) = 1/6 × 1/2 = 1/12.
(b) "Odd number" and "tails" can both happen on the same roll — they're not mutually exclusive. So the simple OR rule doesn't apply here; use a sample space or tree diagram to list paths: (odd, H), (odd, T), (even, T). That's better handled case by case rather than just adding. This is a reminder that the simple addition rule only works when events can't both occur at once.
Check: are they really mutually exclusive?
The most common mistake is adding probabilities when the events can both happen. If events overlap, the simple OR rule overcounts. Check whether both events can occur at once before you decide to add or multiply.
