10 Lecture 9, January 26, 2024

10.0.1 Law of Total Probability

Definition 10.1 (Partition) A sequence of sets \(B_1,B_2,...,B_k\) are said to partition the sample space \(S\) if \(B_i \cap B_j = \emptyset\) for all \(i \ne j\), and \(\cup_{j=1}^k B_j = S\).

Theorem 10.1 (Law of Total Probability) Suppose that \(B_1,B_2,...,B_k\) partition \(S\). Then for any event \(A\), \[ P(A) = P(A | B_1) P(B_1) + P(A | B_2) P(B_2) + \cdots +P(A | B_k) P(B_k). \]

Note: It is a simple usage of LTP such that \[ P(A) = P(A\cap B) + P(A \cap B^c), \] since \(B\) and \(B^c\) partition \(S\) (i.e. \(B\cup B^c = S\) and \(B\cap B^c = \emptyset\))

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10.0.2 Bayes Rule

If we can calculate the conditional profitability, then we may calculate the desire probability by using 1) LTP, 2) definition of conditional probability, and 3) property of the sets (through the Venn diagrams). However, sometimes we have to FLIP the event and the conditioning event. This brings us to the Bayes Theorem.

Theorem 10.2 (Bayes Theorem) Suppose that \(B_1,B_2,...,B_k\) partition \(S\). Then for any event \(A\), \[ P(B_i \mid A ) = \frac{P(A \mid B_i)P(B_i)}{\sum_{j=1}^k P(A \mid B_j) P(B_j) }. \]

This concludes Chapter 4!.