2  Sample Space

Data begins with distinction.

Before a probability can be assigned, before a pattern can be detected, before randomness can even be named, something deeper must already have happened:

the world must have been cut into possibilities.

A die must be allowed to land on one face rather than another.
A measurement must be allowed to take one value rather than another.
A signal must be allowed to rise, fall, persist, or vanish.

This field of possible distinction is called the sample space.

🔰 At first, this may sound technical. It is not. It is one of the most philosophical ideas in all of probability.

A sample space is not merely a list.
It is a decision about what counts as a possible result.

That decision depends on the world, but also on the model.

A coin may be modeled as:

Each of these is a different view of the same event.
The world has not changed.
The space of relevant outcomes has.

⚜️ This is the first lesson:

probability begins not with chance, but with description.

To choose a sample space is to decide what kind of world one is studying.

And once that space is chosen, deeper structures emerge:

From there, probability theory becomes the art of reasoning about what may happen, what must happen, and what can be said before the outcome arrives.

This chapter builds that structure from the ground up.

We begin with a question:

What is the relationship between reality and the spaces we use to describe it?