Chapter I · 18 min

Sample Spaces

From counting to chance

“The most important questions of life are, for the most part, really only problems of probability.”
— Pierre-Simon Laplace, Théorie analytique des probabilités (1812)

You already know how to count. You can list the ways to seat four people, the five-card hands in a deck, the binary strings of length $n$ . Probability begins the moment you decide that some of those lists are lists of things that might happen, and that — for a while — nothing distinguishes one item from another except its name.

If this subject has ever made you feel slow, that was the writing, not you. Every intimidating word in later chapters is a plain idea with a fancy badge. We will pin the badge on only after the idea is sitting in a sentence you could say over chai.

That last clause is the whole classical theory. A fair die is fair because its six faces are interchangeable. A shuffled deck is fair because every ordering is interchangeable. Once you believe that, chance is a ratio of counts.

The sample space

A sample space $Ω$ is the set of mutually exclusive, exhaustive descriptions of how an experiment can come out. Each description is an outcome. We write $ω$ for a typical one.

Exhaustive means: something in $Ω$ will happen. Mutually exclusive means: two of them cannot happen together. If you are sloppy about either, later arithmetic will quietly lie to you.

A fair die as a sample space. The event “even” is the teal subset {2,4,6}. Classical probability is three cells out of six.

For one fair die, $Ω = {1, 2, 3, 4, 5, 6}$ . For two distinguishable dice, $Ω$ has thirty-six ordered pairs. For a coin flipped until heads appears, $Ω = {H, TH, TTH, \dots}$ — already infinite, and already a warning that “count the points” will not be the last word.

The classical rule

An event is a subset $A \subseteq Ω$ . In the classical world, where every outcome is equally likely and $Ω$ is finite,

P (A) = \frac{∣ A ∣}{∣Ω∣} .

The numerator is the count of favourable outcomes; the denominator is the count of possible ones. This is not a theorem. It is a modelling choice: the choice that the labels on the points of $Ω$ carry no extra weight.

Two distinguishable dice. Each of the 36 cells is one outcome. The terracotta diagonal is the event {sum = 7}.

Where counting does the work

The classical rule is only as good as your enumeration. Three habits pay rent for the rest of the book:

Decide whether order matters before you count. Ordered lists and unordered hands are different sample spaces.
Make outcomes equally likely by construction. “A random five-card hand” means each of the $(5 52)$ hands has the same weight. “A random card, then another” is a different experiment.
When a story has stages, write the sample space as a product, or draw a tree. Do not jump to the event.

When equally likely fails

A loaded die still has six faces, but $P ({6})$ is no longer $1/6$ . A thumbtack lands up or down, and nobody gave you a reason to call those two outcomes interchangeable. A waiting time can be any positive real number — there is no “number of points” to divide.

The rest of this book is the repair. We will keep the sample space, keep events as subsets, and replace the ratio-of-counts with a function $P$ that assigns weights. Counting remains the special case in which every singleton weighs the same.

If you want to feel the classical rule with your hands, open the die lab and build events by clicking faces.