139. Total Probability Theorem

Let be a partition of the sample space , meaning:

1. (disjoint)

2. (cover the whole space)

Then for any event A:

The total probability theorem arises by breaking an event into pieces using a partition, and each piece is expressed using the multiplication rule, which itself comes from the definition of conditional probability.

1. Conditional Probability:

2. Multiplication Rule

3. Total Probability Theorem