Event (probabilities)

In Theory of probability, a event is a whole of results of a random test (a subset of the universe). An event being often defined by a proposal, we must be able to say, for any result of the universe, if the event is carried out or not.

Mathematically, an event is an element of a σ-algebra $\backslash \; mathcal\; B$ of a Espace probabilisable $\backslash \; left\; (\backslash \; Omega,\; \backslash \; mathcal\; B\; \backslash \; right)$.

Operations ensemblists on the events

• Is event X following made up of two simple events:

$A\; \backslash \; cup\; B$ (to read “has union B”)

The realization of this event involves the realization of the event has or event B, or of the two events has and B simultaneously.

• Is the event Y following composed of two simple events:

$A\; \backslash \; course\; B$ (to read “has inter B”)

The realization of this event involves the realization of the event has and event B.

• the following relation means that the realization of the event has implies that of the event B automatically.

$A\; \backslash \; subset\; B$ (to read “included in B”)

Expressions random ensemblists of events

Are has, B, C three events.

Crescendo, six probable cases are offered to the experiment:

• No event occurs:

$\backslash \; bar\; has\; \backslash \; \backslash \; course\; \backslash \; \backslash \; bar\; B\; \backslash \; \backslash \; course\; \backslash \; \backslash \; bar\; C\; \backslash$

• Only one event occurs:

• At least an event occurs:

• Two events with the more occur:

• At least two events occurs:

• the three events occur:

(The formulas can beings generalized with a unit NR of events.)

Examples

• the universe Ω is an event, called unquestionable event.
• the empty set is an event, called impossible event.
• for all ω pertaining to Ω the Singleton $\backslash \; \{\backslash \; Omega\; \backslash \}$ is an event, called elementary event.
• We lay out 52 charts and two jokers on a table and we draw only one chart. To draw an individual chart in the universe from the 54 charts, represents a elementary event. But the subsets (including the elementary events) are simply called “events”. Events of this universe can be:
• “to obtain a king” together made up of the 4 kings (union of 4 elementary events),
• “to obtain a chart of heart” (together of 13 charts)
• “to obtain a figure” (together of 12 charts).

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