Probability model

A probability model allows for the formal mathematical description of contingencies. Formally, a probability model is a Measure space $(𝜉, F, ℙ)$ with the additional requirement $ℙ (𝜉) = 1$ , i.e. at least one event must occur. As an overview,

$𝜉$ represents the set of mutually exclusive outcomes (world-states);
$F \subseteq 2 𝜉$ is a σ-algebra of possible events closed under compliment, finite union, and finite intersection; and
$ℙ : F \to [0, 1]$ is the probability measure of an event.

Note in some cases, especially discrete ones, it is unnecessary to limit what kind events are allowed, and so $F = 2 𝜉$ is assumed.

An event here represents some (possibly infinite) union of outcome singletons, i.e. an event is a set of outcomes which would fulfil the event. The σ-algebra contains at least $𝜉$ and $\emptyset$ , and allows for the formation of events from others by

The intersection of events $𝐴 \cap 𝐵$ , which represents the fulfilment of both (and)
The union of events $𝐴 \cup 𝐵$ , which represents the fulfilment of either (or)
The compliment of an event $―― 𝐴$ , which represents the non-fulfilment of $𝐴$ (not)

The probability of any such event is $ℙ (𝐸)$ .

Properties

Some of these follow from measure space Properties

$ℙ (\emptyset) = 0$
$ℙ$ is monotone on $F$ ordered by inclusion, i.e. $𝐴 \subseteq 𝐵 ⟹ ℙ (𝐴) \leq ℙ (𝐵)$ .
For any $𝐴 \in 𝐸$ , it holds that $ℙ (𝐴 𝑐) = 1 - 𝑃 (𝐴)$
$ℙ (𝐴 \cup 𝐵) = ℙ (𝐴) + ℙ (𝐵) - ℙ (𝐴 \cap 𝐵)$

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