Sample Space and Outcomes

Every probability model starts with a sample space — a description of all possible things that may happen during an experiment.

The Experiment

In probability theory, an experiment is any process that produces uncertain outcomes:

Flipping a coin
Rolling a die
Drawing cards from a deck
Measuring the time until a light bulb fails

Sample Space (Ω)

The sample space Ω is the set of all possible outcomes of an experiment.

We sometimes write it as a list — for example, a coin flip has Ω = {Heads, Tails}. But formally it’s a set, and for continuous sample spaces (like all points in a square), we can’t list them at all.

Requirements for a Valid Sample Space

A valid sample space must satisfy two requirements:

Requirement	Meaning
Mutually Exclusive	No two outcomes can occur simultaneously. If one happens, the others cannot.
Collectively Exhaustive	Every possible result is included. Nothing is forgotten.

Discrete vs. Continuous Sample Spaces

Sample spaces come in two fundamental types:

Discrete Sample Spaces

The outcomes can be counted (finite or infinite).

Type	Example	Sample Space
Finite	Coin flip	Ω = {Heads, Tails} — 2 outcomes
Finite	Two coin flips	Ω = {HH, HT, TH, TT} — 4 outcomes
Countably Infinite	Wait until first heads	Ω = {1, 2, 3, 4, …} — infinite but countable

For discrete spaces, you can often list or enumerate the outcomes.

Continuous Sample Spaces

The outcomes are uncountably infinite — there are infinitely many outcomes, and they cannot be arranged in a list.

Experiment	Sample Space
Time until light bulb fails	Ω = [0, ∞) — all non-negative real numbers
Dart thrown at unit square	Ω = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}

For continuous spaces, an outcome is a point in a region. You cannot list the points — you describe the region.

The Dart Example (Continuous)

Experiment: Throw a dart at a square target. You’re skilled enough that the dart always lands inside the square.

Sample Space: All points (x, y) where x and y are real numbers between 0 and 1.

Ω = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}

This is a continuous sample space:

There are infinitely many points in the square
Each point is a possible outcome
You cannot list them all
Any single point has probability zero (a point has zero area)

To assign probabilities, we look at events (subsets with positive area), not individual outcomes.

Examples of Sample Spaces

Experiment	Sample Space	Type
Coin flip	Ω = {Heads, Tails}	Discrete (finite)
Die roll (6-sided)	Ω = {1, 2, 3, 4, 5, 6}	Discrete (finite)
Two coin flips	Ω = {HH, HT, TH, TT}	Discrete (finite)
Wait until first heads	Ω = {1, 2, 3, …}	Discrete (infinite)
Dart in unit square	Ω = {(x,y) : 0≤x,y≤1}	Continuous
Time until bulb fails	Ω = [0, ∞)	Continuous

Choosing the Right Sample Space

Sometimes you have freedom in how detailed to make your sample space.

Example: Flipping a coin, you could use:

Ω₁ = {Heads, Tails}
Ω₂ = {Heads, Tails-and-raining, Tails-and-not-raining}

Both are valid! Choose the level of detail that captures what matters for your model. Irrelevant details (like weather) are usually excluded.

Outcomes

An outcome is a single element of the sample space — one complete result of the experiment.

Properties of Outcomes

Distinguishable — Each outcome is unique and identifiable
Atomic — Cannot be broken down further within this model
Complete — Describes the entire experiment, not just one stage

Outcomes vs. Results

For multi-stage experiments, we distinguish:

Result = what happens at each stage
Outcome = the overall result at the end of the complete experiment

Example: Flip a coin twice

First flip result: Heads
Second flip result: Tails
Outcome: (Heads, Tails) or “HT”

The outcome HT is different from TH — order matters when tracking everything that happens.

Sequential Description (Tree Diagram)

For multi-stage experiments, a tree diagram helps visualize all possible outcomes.