Mutually exclusive events and the OR rule
Maths · Key Stage 3 · The School
What mutually exclusive means
Two events are MUTUALLY EXCLUSIVE when they cannot both happen at the same time. Rolling an even number and rolling an odd number on one die are mutually exclusive — a single roll is one or the other, never both. Being under 18 and being over 65 are mutually exclusive; being a student and being left-handed are not, because both can be true of the same person.
The OR rule for mutually exclusive events
When two events are mutually exclusive, P(A or B) = P(A) + P(B). Rolling a 2 or a 5 on one die: P(2) = 1/6, P(5) = 1/6, so P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3. This works precisely because a single roll cannot give both numbers at once — there is no overlap to double-count.
When simple addition breaks
Drawing a king OR a heart from a pack: these are NOT mutually exclusive, because the king of hearts is both at once. Simply adding P(king) + P(heart) counts the king of hearts twice. The correct approach needs P(A or B) = P(A) + P(B) − P(A and B), subtracting the overlap back out — the same idea that kept a Venn diagram's overlap from being counted twice.
A spinner has these mutually exclusive outcomes: red (0.3), blue (0.25), green (0.2), yellow (0.25) — they sum to 1, since one spin gives exactly one colour. P(red or blue) = 0.3 + 0.25 = 0.55, and this simple addition is valid because a single spin cannot land on two colours at once. Contrast this with a deck of cards, where "spades" and "picture cards" overlap (the picture spades belong to both), so P(spade or picture card) needs the subtraction step. The single question to ask before adding two probabilities is always the same: can both of these happen at once? If no, add them. If yes, subtract the overlap.