# Probability Space Explained: Darts, Events and Area

Throw darts at a board to see a probability space: events are regions, probability is area, and the hit share settles on it. Interactive lab.

## About this tool

A probability space has three parts: a set of possible outcomes, a collection of events you are allowed to ask about, and a rule that gives every event a probability. Here the outcomes are the points of a square board and a dart lands at a uniformly random point. An event is a region you can draw, and its probability is its area. Throw more and more darts and the share that land in a region settles on that area. The same picture shows why probabilities of disjoint events add and why overlapping events need inclusion–exclusion.

## Frequently asked questions

### What is a probability space?

It is the minimal setup for talking about chance: a set Ω of outcomes, a family of events (subsets of Ω you may ask about) and a probability measure P that gives every event a number between 0 and 1. P must give the whole space probability 1 and must add up over disjoint events. Random variables, expectations and everything else in probability are built on these three objects. On this page Ω is the unit square, the events are the regions you draw, and P is area.

### Why is probability the same as area here?

Because the dart is thrown uniformly: no part of the board is favoured, so a region half as big is hit half as often. Area is also additive. Cut a region into pieces that do not overlap and their areas add up to the total, exactly as the probabilities of disjoint events must. That is why a uniform dart turns geometry into probability. Aim at the centre instead and probability stops matching area; you would need a different measure P on the same board.

### What should I try on the page?

Draw two events that overlap and compare the dart frequency of each with its area. Then look at the union: the area of A or B is the area of A plus the area of B minus the area of both, which is inclusion–exclusion. Slide the events apart until they are disjoint and the overlap term vanishes. Throw a few hundred darts, then thousands, and watch the hit share wobble at first and then settle on the exact area.

### A line has area zero. Can a dart land on it?

A dart always lands somewhere, so it lands on a point, and that point lies on many lines. Yet for any one line fixed in advance the chance is exactly zero, because a line has area 0. Probability zero does not mean impossible; it means almost surely not. With continuous outcomes single points carry no probability and only regions do. The idea returns on the Lebesgue measure page.

## Learn the theory

- [Probability spaces and Kolmogorov's axioms](https://quantsandbox.dev/guide/probability/probability-spaces-axioms): Probability as a measure of total mass one: outcomes, events and axioms, discrete and continuous models, infinite coin tossing, almost sure events.
- [Measure theory in option pricing](https://quantsandbox.dev/guide/probability/measure-theory-pricing): How the measure theory and probability of this track meet in option pricing: equivalent measures and densities, information, conditional expectation, limits.
- [Probability spaces, σ-algebras and random variables](https://quantsandbox.dev/guide/stochastic/probability-spaces): Sample space, σ-algebra, probability measure, measurable maps and the σ-algebra generated by a variable.

Open the interactive tool: https://quantsandbox.dev/probability-space