Probability Function¶

Definition¶

Let $S$ be a sample space and let $\mathcal{F}$ be a sigma-algebra of events, that is, a collection of events containing $S$ and closed under complements and countable unions. A probability function is a function $\mathbb{P}\colon\mathcal{F}\to[0,1]$ that assigns probabilities to events and satisfies the axioms of probability.

Interpretation¶

A probability function is the rule that assigns to each event a number between $0$ and $1$ saying how likely it is. Together with the sample space and the collection of events, it forms the probability model.

Events of a die roll, from the empty set to the whole sample space, with arrows to their probabilities on a scale from 0 to 1.

The figure maps events of a fair die roll to $[0,1]$ : $\varnothing\mapsto0$ $0$ , $\{6\}\mapsto\tfrac{1}{6}$ $\frac{1}{6}$ , $\{2,4,6\}\mapsto\tfrac{1}{2}$ $\frac{1}{2}$ , $\{1,3,6\}\mapsto\tfrac{1}{2}$ and $S\mapsto1$ $1$ . The event $\{1,3,6\}$ is the union of the disjoint events $\{1,3\}$ and $\{6\}$ , so its probability is $\tfrac{1}{3}+\tfrac{1}{6}=\tfrac{1}{2}$ $\frac{1}{2}$ . The events $\{2,4,6\}$ and $\{1,3,6\}$ show that different events can have the same probability: the function need not be injective.

The input of $\mathbb{P}$ is an event, a set of outcomes, not an outcome: strictly one writes $\mathbb{P}(\{6\})$ , and $\mathbb{P}(6)$ is a common shorthand. The axioms restrict which functions are allowed, but they do not choose one; that is a modelling decision. A fair die and a loaded die have the same sample space and the same events, but different probability functions. On a finite sample space, the probability function is determined by the probabilities of the single outcomes, since by additivity $\mathbb{P}(A)$ is the sum of $\mathbb{P}(\{\omega\})$ over the outcomes $\omega\in A$ $A$ .

Naive probability is the special case in which all single outcomes get the same probability. Conditioning on an event $B$ with $\mathbb{P}(B)>0$ produces a new probability function, $A\mapsto\mathbb{P}(A\mid B)$ (conditional probability).

Key Results¶

A probability function assigns numbers to events according to the probability axioms.
The full sample space has probability one.
Disjoint unions are handled by adding probabilities.

Examples¶

For a fair die, the probability function assigns probability one sixth to each single-face event.
In a reliability model, the probability function assigns a probability to the event that the system fails.
For a deck of cards, the probability function assigns probabilities to events such as drawing a heart.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 2

Aliases: probability measure Keywords: probability Symbols: element of, omega

Last updated: 20 October 2026 03:02