1. The Foundation: Partitions (Complete System of Hypotheses)
Big Idea: Before any formula, you must learn to see the partition in a story. Every problem in this topic begins with the same structural question: "What are the mutually exclusive, exhaustive hypotheses?"
Definition — Partition of a Sample Space
A collection of events $H_1, H_2, \ldots, H_n$ is called a partition of the sample space $\Omega$ if:
- Pairwise incompatible (mutually exclusive): $H_i \cap H_j = \emptyset$ for all $i \neq j$ — no two hypotheses can occur simultaneously.
- Exhaustive (cover the whole space): $H_1 \cup H_2 \cup \cdots \cup H_n = \Omega$ — one of them must occur.
- Positive probabilities: $P(H_i) > 0$ for every $i$.
Consequence: the weights sum to one, $\sum_{i=1}^{n} P(H_i) = 1$. This is your first sanity check in every problem.
Recognizing Partitions in Stories — The Modeling Skill
The real skill is not plugging into a formula — it is reading a word problem and identifying the partition. Here are the classic patterns: