Multiplication Rule (Probability)
Statistics & ProbabilityThe 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.
Example
Probability of flipping heads AND rolling a $3$ (independent events): $P = 1/2 \times 1/6 = 1/12$.
Key Insight
"And" in probability usually means multiply. "Or" usually means add (with subtraction for overlap).
Definition
The multiplication rule: $P(A \text{ and } B) = P(A)P(B|A)$. For independent events, $P(B|A) = P(B)$, so $P(A \text{ and } B) = P(A)P(B)$. For dependent events, you must use the conditional probability of $B$ given that $A$ occurred.
Example
Drawing $2$ aces without replacement: $P(\text{first ace}) = 4/52$. Given first ace drawn, $P(\text{second ace}) = 3/51$. $P(\text{both aces}) = 4/52 \times 3/51 = 12/2652 = 1/221$.
Key Insight
The multiplication rule can be extended: $P(A \text{ and } B \text{ and } C) = P(A)P(B|A)P(C|A \text{ and } B)$. For independent events, this simplifies to $P(A)P(B)P(C)$.
Definition
The chain rule of probability: $P(A_1 \cap A_2 \cap \ldots \cap A_n) = P(A_1)P(A_2|A_1)P(A_3|A_1 \cap A_2) \cdots P(A_n|A_1 \cap \ldots \cap A_{n-1})$. For independent events, all conditional probabilities equal the unconditional ones, and the product simplifies to the product of individual probabilities.
Example
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. This exponentially reduces the number of parameters needed to specify the full joint distribution.
Key Insight
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.