The Addition Rule for Combined Events
Simple Explanation
To find the probability that A or B happens (or both), add their individual probabilities and subtract the probability that both happen, to avoid double-counting: P(A∪B) = P(A) + P(B) − P(A∩B). If A and B can never happen together (mutually exclusive), P(A∩B) = 0, simplifying it to just P(A) + P(B).
Why Do We Need It?
This rule correctly combines the probabilities of two events without over-counting outcomes that belong to both — a mistake that is easy to make if you simply add probabilities without checking for overlap.
Formula
The Addition Rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A, B are mutually exclusive: P(A ∪ B) = P(A) + P(B).
The probability that A or B happens equals the sum of their individual probabilities, minus the probability they both happen (to avoid double-counting the overlap).
- P(A ∪ B)
- — the probability that A happens, or B happens, or both
- P(A ∩ B)
- — the probability that both A and B happen
- mutually exclusive
- — A and B cannot both happen at once, so P(A ∩ B) = 0
When to use it: Whenever you need the probability that at least one of two events happens.
Worked Example
Apply the addition rule with overlap
A card is drawn from a standard deck. Find P(king or heart).
Why Does This Work?
Simply adding P(A) + P(B) counts every outcome in the overlap A∩B twice — once as part of A, once as part of B — so subtracting P(A∩B) once corrects the count back to exactly right.
Real-Life Example
Membership in two clubs
A school wants to know the probability that a randomly chosen student is in the chess club or the debate club, when some students are in both.
Simply adding the two clubs' membership fractions would double-count students in both clubs — the addition rule's subtraction step corrects for exactly that overlap.
Practice
P(A) = 0.5, P(B) = 0.3, and A, B are mutually exclusive. Find P(A or B).
MediumP(A) = 0.6, P(B) = 0.4, P(A∩B) = 0.2. Find P(A or B).
MediumCommon mistake
Always subtracting P(A∩B) even when the events are mutually exclusive (where it is already zero) is harmless, but forgetting to subtract it when events DO overlap leads to an answer that is too high.
Quick Review
- P(A∪B) = P(A) + P(B) − P(A∩B) in general.
- Mutually exclusive events: P(A∩B) = 0, so P(A∪B) = P(A) + P(B).
- Always check for overlap before simply adding two probabilities.