Multiplication Rule (Probability)
The multiplication rule gives the probability that two events both occur, using conditional probability for dependent events.
Formula
P(A \text{ and } B) = P(A) \times P(B|A)
Definition
The multiplication rule tells you the probability of two events both happening: for independent events, just multiply their individual probabilities. Formally, $P(A \text{ and } B) = P(A)P(B|A)$; for independent events, $P(B|A) = P(B)$, so $P(A \text{ and } B) = P(A)P(B)$, while for dependent events you must use the conditional probability of $B$ given that $A$ occurred. The chain rule of probability generalizes this: $P(A_1 \cap A_2 \cap \ldots \cap A_n) = P(A_1)P(A_2|A_1)P(A_3|A_1 \cap A_2) \cdots P(A_n|A_1 \cap \ldots \cap A_{n-1})$, which simplifies to the product of individual probabilities when all events are independent.
Example
The probability of flipping heads AND rolling a $3$ (independent events) is $P = 1/2 \times 1/6 = 1/12$. Drawing $2$ aces without replacement: $P(\text{first ace}) = 4/52$, and given the first ace was drawn, $P(\text{second ace}) = 3/51$, so $P(\text{both aces}) = 4/52 \times 3/51 = 1/221$. In Bayesian networks, the chain rule factorizes the joint distribution over all variables as a product of conditional distributions, one per variable given its parents in the DAG, exponentially reducing the number of parameters needed to specify the full joint distribution.
Key Insight
"And" in probability usually means multiply, while "or" usually means add (with subtraction for overlap); the multiplication rule extends naturally, $P(A \text{ and } B \text{ and } C) = P(A)P(B|A)P(C|A \text{ and } B)$, simplifying to $P(A)P(B)P(C)$ for independent events. The chain rule is the foundation of Bayesian network factorization: a joint distribution $P(X_1, \ldots, X_n)$ requires $O(2^n)$ parameters in general, but a Bayesian network with sparse parent structure needs only $O(n \cdot 2^k)$ parameters, where $k$ is the maximum number of parents per node.