# Probability space

## In words...

A probability space is a triplet made of a sample space, a set of events and a probability measure.

The sample space is the set of all possible outcomes of a given random experiment.

The set of events contains events, where each event is itself a set of outcomes. The set of all possible events must be closed under countably many complement, union and intersection operations. For instance, this means that the union of two events must also belong to the set of events.

The probability measure is a function that assigns a probability (a number between 0 and 1) to any event from the set of events. In addition, a probability measure must satisfy other axioms. In particular, the probability of the empty set must be 0 while the one of the entire sample space must be 1. Another requirement is that the probability of a union of disjoint events equals the sum of their probabilities.

A probability measure enjoys a number of basic properties.

## In picture...

### A simple random experiment

Consider a simple experiment in which we pick at random one out of three people.

The sample space is the set of three people: $\Omega = \{Teddy, Bob, John\}$.

The set of all possible events is
$\Sigma = \{$
 $\emptyset$ (no one), $\{Teddy\}$, $\{Bob\}$, $\{John\}$ , $\{Teddy, Bob\}$, $\{Teddy, John\}$, $\{Bob, John\}$, $\{Teddy, Bob, John\}$ (everyone),
$\}$

For each of these events, the probability measure $P$ assigns a probability between 0 and 1. Assume that the probabilities of picking each individual are fixed: , and . Then, we can compute all the other probabilities from these ones.

Hover over the list of events above to compute their probabilities

## In maths...

Consider a random experiment and let $\Omega$ denote the set of all possible outcomes, which we call the sample space.
We call an event any set $A$ such that $$A = \bigcup_{i=1}^n \{\omega_i\}, \quad \omega_i\in\Omega, \ n\in\{0,\dots,|\Omega|\} .$$ Define the set of events $\Sigma$ as a $\sigma$-algebra on $\Omega$, i.e., a set of events (such as $A$ above) that is closed under countably many complement, union and intersection operations.
Finally, introduce a probability measure $P$, i.e., a function assigning a real number to any event, $$P : \Sigma \rightarrow [0,1] ,$$ such that $$P(\emptyset) = 0,$$ $$P(\Omega) = 1 ,$$ and, for disjoint events $A_i$ (meaning that for $i\neq j$, $A_i\cap A_j = \emptyset$ ), $$P\left( \bigcup_{i=1}^\infty A_i \right) = \sum_{i=1}^\infty P(A_i) .$$

Then, the triplet $(\Omega, \Sigma, P)$ is a probability space.