📐 The Total Probability Formula
A powerful tool for computing the probability of an event by breaking it down into simpler, mutually exclusive scenarios.
🔑 Prerequisite Concepts
- Conditional Probability: $P(A|B) = \dfrac{P(A \cap B)}{P(B)}$, the probability of $A$ given that $B$ has occurred.
- Partition of Sample Space: Events $B_1, B_2, \ldots, B_n$ form a partition of the sample space $S$ if:
- They are mutually exclusive: $B_i \cap B_j = \emptyset$ for all $i \neq j$
- They are exhaustive: $B_1 \cup B_2 \cup \cdots \cup B_n = S$
- Each has positive probability: $P(B_i) > 0$
📜 Formal Definition
Theorem: Law of Total Probability
If $B_1, B_2, \ldots, B_n$ form a partition of the sample space $S$, then for any event $A$:
$$P(A) = \sum_{i=1}^{n} P(A|B_i) \cdot P(B_i)$$
Expanded form:
$$P(A) = P(A|B_1)P(B_1) + P(A|B_2)P(B_2) + \cdots + P(A|B_n)P(B_n)$$