3  Lecture 2: Conditional Probability and Bayes’ Rule

3.1 Conditional Probability

The probability of ( A ) given ( B ) is:

\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \]

3.2 Bayes’ Rule

\[ P(B_j \mid A) = \frac{P(A \mid B_j) P(B_j)}{\sum_k P(A \mid B_k) P(B_k)} \]

This allows us to reverse conditional probabilities based on known likelihoods and priors.