140. Total Probability Theorem

Let ${𝐵}_1,{𝐵}_2,\dots ,{𝐵}_{𝑛}$ be a partition of the sample space $\mathrm{\Omega }$, meaning:

1. ${𝐵}_{𝑖}\cap {𝐵}_{𝑗}=\varnothing \forall 𝑖\ne 𝑗$ (disjoint)

2. ${\bigcap }_{𝑖=1}^{𝑛}{𝐵}_{𝑖}=\mathrm{\Omega }$ (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