is a type of mathematical space
is a type of measure space such that the measure of the whole space is equal to one
in probability theory, a probability space or a probability triple (𝛺, 𝐹, 𝐏) is a mathematical construct that provides a formal model of a random process:
- a sample space/set 𝛺 which is the set of all possible outcomes
- an event space/set 𝐹 more specifically a σ-algebra on 𝛺, where each event is a subset of 𝛺 i.e. a set containing zero or more outcomes (there are a maximum of 2^|𝛺| events)
- a probability measure 𝐏: 𝐹 → [0, 1] that assigns each event in the event space a probability (i.e. a number between 0 and 1)

Probability Space - Definition

A probability space is a triple (𝛺, 𝐹, 𝑃) consisting of:

a sample space 𝛺 is the non-empty set of all possible outcomes
the event space σ-algebra 𝐹⊆2^𝛺 (also called σ-field) – a set of subsets of 𝛺, called events, such that:
- 𝐹 contains the entire sample space: 𝛺∊𝐹
- 𝐹 is closed under complements: if 𝐴∊𝐹, then also 𝛺\𝐴∊𝐹
- 𝐹 is closed under countable unions: if 𝐴_𝑖∊ 𝐹 for 𝑖=1,2,…, then also (⋃_{1≤𝑖≤∞} 𝐴_𝑖) ∊ 𝐹
- 𝐹 is closed under countable intersections: if 𝐴_𝑖∊ 𝐹 for 𝑖=1,2,…, then also (⋂_{1≤𝑖≤∞} 𝐴_𝑖) ∊ 𝐹 # the corollary from the previous two properties and De Morgan’s law
the probability measure, 𝐏: 𝐹 → [0, 1], that assigns each event in the event space a probability, which is a number between 0 and 1
- 𝐏 is countably additive (also called σ-additive): if {𝐴_𝑖}_{1≤𝑖≤∞} ⊆ 𝐹 is a countable collection of pairwise disjoint sets, then 𝐏(⋃_{1≤𝑖≤∞} 𝐴_𝑖) = 𝛴_{1≤𝑖≤∞} 𝐏(𝐴_𝑖)
- the measure of the entire sample space is equal to one: 𝐏(𝛺) = 1

Probability Space - Example of Flipping 2 Coins

𝛺 = {(H,H), (H,T), (T,H), (T,T)}

for example: