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
- \(0 \le P(A \mid B) \le 1\).
This follows from the fact that if \(A \subset B\) then \(P(A) \le P(B)\)
\(P(A^c \mid B) = 1-P(A \mid B)\).
If \(A_1\) and \(A_2\) are disjoint (i.e. \(P(A_1\cap A_2)=\emptyset\): \(P(A_1 \cup A_2 \mid B) = P(A_1 \mid B) + P(A_2 \mid B)\).
\(P(S \mid B)= 1 = P(B \mid B)\).
Definition 9.3 (Product rule) For any events \(A\) and \(B\), we have \[ P(A\cap B) = P(A\mid B) P(B) = P(B \mid A) P(A). \]