Multiplication Rule for Conjunct

Multiplication Rule

$P (A \cap B) = P (A) \cdot P (B | A)$

Sports Car Example

Say 10% of all new vehicles purchased from a particular dealer are sports cars. Of those, 30% have heated seats.
Consider randomly selecting someone who purchased a vehicle from this dealer. Define $A$ as the the event where the individual selected a sports car, and $B$ as the event of selecting heated seats. Then $P (A) = .10$ and $P (B | A) = .30$ . The probability that they got a sports car with heated seats is:

P (A \cap B) = P (A) \cdot P (B | A) = .10 \cdot .30 = .03

The rule can be used for multiple conjunctions:

P (A \cap B \cap C) = P (C | A \cap B) \cdot P (A \cap B) = P (C | A \cap B) \cdot P (B | A) \cdot P (A)

Note

The Multiplication Rule for Conjunct is just derived from the Definition of Conditional Probability definition.