Probability Space

TL;DR

A probability space formalizes all possible outcomes of a random experiment and how likely events are.
It comprises three parts: a sample space (all outcomes), a set of events (subsets of outcomes), and a probability measure (assigns probabilities).
Used to compute probabilities and make predictions about random outcomes.

Definition

A probability space is a mathematical construct used in probability theory to represent the set of outcomes of a random event or experiment. It consists of three components: a sample space, which is the set of all possible outcomes; a set of events, which are subsets of the sample space; and a probability measure, which assigns a probability to each event.

Explanation

The sample space lists every possible outcome of the experiment.
Events are defined as subsets of that sample space; each event contains one or more outcomes.
The probability measure maps each event to a probability value, allowing calculation of the likelihood of events by aggregating the probabilities of the outcomes they contain.
Probability spaces apply to discrete examples (finite sets of outcomes), combinatorial examples (such as shuffled decks), and continuous or infinite-outcome scenarios.

Examples

Roll of a die

Sample space: {1, 2, 3, 4, 5, 6}
Example events: “rolling an even number” = {2, 4, 6}; “rolling a number greater than 3” = {4, 5, 6}
Example probabilities: the event “rolling an even number” has probability 1/2; the event “rolling a number greater than 3” has probability 1/3.

Flip of a coin

Sample space: {heads, tails}
Example events: “flipping heads” = {heads}; “flipping tails” = {tails}
Example probabilities: each of these events has probability 1/2 for a fair coin.

Deck of cards (shuffling)

Sample space: all possible arrangements of the cards in the deck
Example events: “drawing a specific card” or “drawing a specific combination of cards”
The probability measure assigns probabilities based on the likelihood of the outcomes in each event.

Continuous / infinite outcomes

Example: rolling a dice with an infinite number of sides
Sample space: all possible outcomes on the infinite number of sides
Example events: “rolling a number between 1 and 3” or “rolling a number greater than 5”
The probability measure assigns probabilities based on the likelihood of the outcomes in each event.

Use cases

Representing the outcomes of random events and experiments.
Assigning probabilities to events to calculate the likelihood of specific outcomes.
Making predictions about which outcomes are more or less likely.

Sample space
Event
Probability measure
Random event
Experiment