Multiplication theorem and independent events

The essentials

Rearranging the definition of conditional probability gives the multiplication theorem: P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B). It is used when events happen in sequence, such as drawing two balls one after the other without replacement. For three events, P(A ∩ B ∩ C) = P(A) P(B|A) P(C|A ∩ B).

Two events are independent if the occurrence of one does not change the probability of the other, that is, P(B|A) = P(B). Then the multiplication theorem simplifies to P(A ∩ B) = P(A) × P(B). Draws with replacement and separate tosses of coins or dice give independent events.

Key points

• If A and B are independent, so are A and B′, A′ and B, and A′ and B′.
• For independent events, P(A ∪ B) = P(A) + P(B) − P(A)P(B).
• P(at least one of A, B, C) = 1 − P(A′)P(B′)P(C′) for independent events.
• Mutually exclusive means P(A ∩ B) = 0; independent means P(A ∩ B) = P(A)P(B). If both P(A) and P(B) are positive, the two conditions cannot hold together.
• Three events are mutually independent only if every pair and the triple satisfy the product rule.

Exam tips

• "Without replacement" changes the second probability; "with replacement" does not.
• For "at least one" problems, work with the complement.

Common mistakes

• Treating mutually exclusive events as independent.
• Multiplying probabilities for dependent events without adjusting the second one.