collectively exhaustive events

A collection of events is collectively exhaustive when, taken together, they cover every possible outcome — at least one of them is guaranteed to happen on any run of the experiment. As sets, their union equals the whole sample space: A1 ∪ A2 ∪ ... = Omega. For one die, the events 'is 3 or less' and 'is 4 or more' are collectively exhaustive, since every face falls into at least one of them; so is the larger list {1}, {2}, {3}, {4}, {5}, {6}. Exhaustive is about leaving nothing out.

This idea pairs naturally with mutual exclusivity, but the two are independent properties. Exhaustive says the events cover everything; mutually exclusive says they never overlap. When a collection is both at once — covering everything with no overlaps — it is called a partition of the sample space, and partitions are the workhorse behind the law of total probability you meet later. The simplest partition of all is an event A together with its complement A^c: between them they cover the whole space and never overlap.

Because some event in an exhaustive collection always occurs, their probabilities must add up to at least 1. If the collection is also mutually exclusive (a partition), the probabilities add up to exactly 1 — no double counting, nothing missed. This is why dividing a problem into a partition is such a reliable strategy: you can compute the chance of a target event by handling each piece of the partition separately and summing, confident that the pieces together account for the whole of certainty.