Addition Theorem of Probability: Explained with Examples

The addition theorem of probability answers the most common or-question in the subject: what is the probability that at least one of two events happens? The answer is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). You add the individual probabilities, then subtract the overlap so outcomes belonging to both events are not counted twice. When A and B are mutually exclusive, the overlap is empty and the formula simplifies to P(A ∪ B) = P(A) + P(B).

This theorem matters because plain addition is the instinct almost everyone has, and it is wrong whenever the events overlap. Drawing a card that is a king or a spade is not 4/52 + 13/52, because the king of spades was counted in both groups. The subtraction term is the correction that makes the instinct exact.

Key Concepts

• Union A ∪ B: the event that A or B or both occur, read as A or B.
• Intersection A ∩ B: the event that both occur; its probability is the correction term subtracted.
• Mutually exclusive case: A ∩ B is empty, so P(A ∩ B) = 0 and the theorem reduces to simple addition.
• Inclusion-exclusion principle: the pattern of adding singles, subtracting pairs, adding triples, shown by the three-event formula.
• Assumptions: none beyond the probability axioms; the theorem holds for every pair of events.

Formulas

For any two events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events: P(A ∪ B) = P(A) + P(B). These describe the union only; the multiplication theorem handles the intersection.

Worked Example

One card is drawn from a 52-card deck. Find the probability it is a king or a spade. Let A be drawing a king and B drawing a spade. P(A) = 4/52 and P(B) = 13/52. The overlap A ∩ B is the king of spades, one card, so P(A ∩ B) = 1/52. By the addition theorem, P(A ∪ B) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13. Direct counting confirms: 4 kings plus 12 non-king spades = 16 favourable cards out of 52. Naive addition would give 17/52, overcounting the king of spades.

Common Misconceptions

• Always add for or-questions: Only for mutually exclusive events. For overlapping events, skipping the subtraction overcounts and can even give probabilities above 1.
• The subtraction term is P(A) × P(B): That product equals P(A ∩ B) only when A and B are independent. In general you need the actual intersection probability.
• The three-event formula is just adding three probabilities: It needs three pair-subtractions and one triple-addition; leaving them out is the same double-counting mistake at larger scale.

Real-World Applications

The addition theorem is used whenever at least one of several risks or successes matters: an engineer computing the chance that at least one backup system works, a doctor estimating the chance a patient has at least one of two conditions, or a marketer finding the fraction reached by at least one of two ad campaigns without double-counting those reached by both.

The addition theorem states P(A ∪ B) = P(A) + P(B) − P(A ∩ B), collapsing to simple addition for mutually exclusive events. The subtracted intersection corrects the double counting of naive addition. Extended by inclusion-exclusion to three or more events, it is the universal tool for every or-question in probability.