Sample Spaces and Events

Almost every probability mistake is, at root, a mistake about the sample space. Getting the space right first makes the rest of the calculation mechanical, so this lesson is about the discipline of writing down exactly what can happen before asking how likely anything is.

Outcomes and events

The distinction between outcome and event is worth dwelling on. An outcome is one complete result of the experiment; an event is a collection of outcomes described by some property. For one die, , the outcome of a particular roll might be , and the event "the roll is even" is the subset , which occurred because belongs to it. Single outcomes, whole subsets, the empty set, and itself all count as events; the last two are the events that never happen and always happen respectively.

Choosing the right space

For two dice the natural-looking sample space of eleven possible sums, through , is a trap: those sums are genuinely the possible results, but they are not equally likely, so the classical definition cannot be applied to them directly. The equally likely outcomes are the ordered pairs , of which there are . The right sample space is the one whose atoms are interchangeable by symmetry, even if it is larger and records more detail than the question asks about.

Keeping the dice ordered even when they are physically identical often puzzles people. The resolution is that "identical" describes the dice, not the outcomes: rolling and rolling are different results of the experiment even if you cannot tell the dice apart, and each is as likely as . Distinguishability is a property of your bookkeeping, and the bookkeeping that keeps outcomes equally likely is the one to use.

Larger and infinite spaces

Sample spaces grow quickly. Five coins give sequences; a shuffled deck has orderings, a number with 68 digits. Nobody lists such spaces, but the discipline survives: you reason about the space's structure and count within it. Sample spaces can also be infinite. "Flip until the first head" has outcomes H, TH, TTH, and so on without end, and "pick a random point on a line segment" has a continuum of outcomes, where single points get probability zero and only intervals carry probability. Both kinds appear later in this course under discrete and continuous random variables.

Writing out the sample space feels slow, and that is the point. The candidates who pause to fix the space are the ones who compute quickly and correctly afterwards, because every subsequent step is checked against a concrete set rather than a vague intuition.