Probability theory MOC

Probability model

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

• represents the set of mutually exclusive outcomes (world-states);
• is a σ-algebra of possible events closed under compliment, finite union, and finite intersection; and
• 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 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 and , and allows for the formation of events from others by

• The intersection of events , which represents the fulfilment of both (and)
• The union of events , which represents the fulfilment of either (or)
• The compliment of an event , which represents the non-fulfilment of (not)

The probability of any such event is .

Properties

Some of these follow from measure space Properties

2. is monotone on ordered by inclusion, i.e. .
3. For any , it holds that

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