a partition of the sample space
A partition is a way of slicing the set of all possible outcomes into separate, non-overlapping pieces that together leave nothing out. Think of cutting a pizza: every point of the pizza belongs to exactly one slice, no slice overlaps another, and the slices together make the whole pie. In probability the 'pizza' is the sample space and the 'slices' are events.
Formally, events B1, B2, ... Bk form a partition when they are pairwise mutually exclusive (no two can happen together, Bi and Bj overlap is empty for i not equal j) and collectively exhaustive (their union is the entire sample space, so one of them must occur). The cleanest partition has just two pieces: an event B and its complement not B. Either B happens or it does not — that always partitions the world. A die can be partitioned by 'low (1-2), medium (3-4), high (5-6)', or by 'even versus odd', and so on.
Partitions are the scaffolding for case-based reasoning. The law of total probability and Bayes' theorem both require a partition so that the case probabilities P(Bi) add up to exactly 1 — the weights in the weighted average. If you pick cases that overlap, an outcome gets counted in two slices and your probabilities exceed 1; if your cases leave a gap, some outcomes are never counted and you lose probability. Choosing the RIGHT partition — the one where the conditional probabilities are easy to know — is often the whole art of a problem.
To analyze a quiz show, partition contestants by the prize door they chose: door 1, door 2, door 3. These three events are mutually exclusive (you pick one door) and exhaustive (you must pick some door), so P(door 1) + P(door 2) + P(door 3) = 1, a valid partition for total-probability calculations.
Mutually exclusive plus exhaustive equals a partition; the case probabilities sum to 1.
Mutually exclusive alone is not enough, and exhaustive alone is not enough — a partition needs both. 'Sick' and 'over 60' do not partition a population; 'sick' and 'not sick' do.