1. **Problem Statement:** Prove the Law of Total Probability which states that if $A_1, A_2, ..., A_n$ are mutually exclusive and exhaustive events with $P(A_i) > 0$ for $i=1,2,...,n$, and $B$ is any event, then $$P(B) = \sum_{i=1}^n P(A_i) \times P(B|A_i).$$ 2. **Step 1: Understand the setup** - Events $A_1, A_2, ..., A_n$ are mutually exclusive which means $A_i \cap A_j = \emptyset$ for $i \neq j$. - They are exhaustive, so their union covers the entire sample space: $\bigcup_{i=1}^n A_i = S$. 3. **Step 2: Use the Multiplication Rule** The probability of the event $B$ can be decomposed as the sum of probabilities of intersections with each $A_i$ because $A_i$ are mutually exclusive: $$P(B) = P(B \cap A_1) + P(B \cap A_2) + \cdots + P(B \cap A_n).$$ 4. **Step 3: Apply Conditional Probability Definition** Recall the definition of conditional probability: $$P(B|A_i) = \frac{P(B \cap A_i)}{P(A_i)} \quad \Rightarrow \quad P(B \cap A_i) = P(A_i) \times P(B|A_i).$$ 5. **Step 4: Substitute back into sum** Substituting, we get: $$P(B) = \sum_{i=1}^n P(A_i) \times P(B|A_i).$$ 6. **Step 5: Interpret the rectangular graph** - The vertical sections labeled $A_1, A_2, ..., A_n$ represent the events partitioning the sample space. - The horizontal section labeled $B$ shows the event whose probability we want to find. - The intersection areas correspond to $P(B \cap A_i)$, illustrating how $B$ overlaps each $A_i$ region. This visual reinforces the idea that total probability of $B$ is the sum of its intersections with all $A_i$. **Final answer:** $$P(B) = \sum_{i=1}^n P(A_i) \times P(B|A_i).$$