Read Chapter 1.4 of the text.

Points to pay attention to:

• We have already used the law of total probability multiple times in the problems we have done in the Conditional Probability Module. It is not a new concept, but simply an observation that the following concepts are often used together:
  • the definition of conditional probabilities: \(P(A_i \textrm{ and } B) =P(A_i\cap B)= P(B \mid A_i)P(A_i)\), and
  • probability of a disjoint union is the sum of probabilities of the components of the union: \(P( \cup_i (A_i \cap B) ) = \sum_i P(A_i\cap B)\) for disjoint \(A_i\).
    Can you pinpoint all uses in the problems of prior modules?
• Bayes theorem seems simple as a standalone formula. But it is not just a formula we are trying to learn, but a way of thinking. Pay attention to how this changes the formulation of the problems we solve in class in this module.