Addition Rule for Probabilities
The addition rule for probabilities describes how to calculate the probability that at least one of two events occurs. There are two forms: one for mutually exclusive events (no overlap) and a general form that accounts for overlap.
Formulas
-
For mutually exclusive events Y and Z:
P(Y or Z) = P(Y) + P(Z) -
For non-mutually exclusive events Y and Z:
P(Y or Z) = P(Y) + P(Z) − P(Y and Z)
The second formula corrects for double-counting the overlap P(Y and Z). The mutually exclusive case is a special case of the general formula where P(Y and Z) = 0.
Explore More Resources
Examples
-
Die roll (mutually exclusive):
Probability of rolling a 3 or a 6 = 1/6 + 1/6 = 2/6 = 1/3.
You cannot roll both a 3 and a 6 on a single roll, so the events are mutually exclusive. -
Class selection (non-mutually exclusive):
Class: 9 boys and 11 girls (20 students). B grades: 4 boys and 5 girls (9 students).
P(girl) = 11/20, P(B) = 9/20, P(girl and B) = 5/20.
P(girl or B) = 11/20 + 9/20 − 5/20 = 15/20 = 3/4.
Mutual Exclusivity (brief)
Mutually exclusive events cannot occur at the same time. If two events are mutually exclusive, the probability that both occur is zero.
Key Points
- Use the general formula P(Y or Z) = P(Y) + P(Z) − P(Y and Z) in most situations to avoid double-counting.
- If events are mutually exclusive, the overlap term is zero and the rule simplifies to P(Y or Z) = P(Y) + P(Z).
- Distinguish mutual exclusivity (cannot occur together) from independence (occurrence of one does not affect the probability of the other).