9 Lecture 8, January 24, 2024

Definition 9.1 (Conditional Probability) The conditional probability of an event $A$ given an event $B$, assuming $P(B)>0$, is

$$P(A \mid B) = \frac{P(A\cap B)}{P(B)}.$$

Definition 9.2 (Equivalent definition of independence) Two events $A$ and $B$ are independent, if $$P(A|B)=P(A),$$ provided $P(B)>0$.

9.0.1 Properties of Conditional Probability

1. $0 \le P(A \mid B) \le 1$.

This follows from the fact that if $A \subset B$ then $P(A) \le P(B)$

2. $P(A^c \mid B) = 1-P(A \mid B)$.

3. If $A_1$ and $A_2$ are disjoint (i.e. $P(A_1\cap A_2)=\emptyset$: $P(A_1 \cup A_2 \mid B) = P(A_1 \mid B) + P(A_2 \mid B)$.

4. $P(S \mid B)= 1 = P(B \mid B)$.

Definition 9.3 (Product rule) For any events $A$ and $B$, we have $$P(A\cap B) = P(A\mid B) P(B) = P(B \mid A) P(A).$$