Axioms of the probabilities

In the Theory of probability, a Probabilité $\backslash \; P$ is an application which with a event $\backslash \; unspecified\; A$ related to the random Expérience $\backslash \; mathcal\; E$ associates a real number (noted $\backslash \; P\; (A)$) definite in such a way that it satisfies the Axiomes of the probabilities or axioms of Kolmogorov , of the name of Andrei Nikolaievitch Kolmogorov, mathematician Russian who developed them

First axiom

For any event $\backslash \; A$:

$0\; \backslash \; Leq\; P\; (A)\; \backslash \; Leq\; 1.$

I.e. the probability of an event is represented by a real number ranging between 0 and 1.

Second axiom

$\backslash \; \backslash \; Omega$ indicating the universe associated with the random experiment considered,

$\backslash \; P\; (\backslash \; Omega)\; =\; 1$,

I.e. the probability of the unquestionable event, or to obtain any result of the universe, is equal to 1. In other words, the probability of carrying out one or the other of the elementary events is equal to 1.

Third axiom

Any continuation of disjoined events two to two (one also says: two to two incompatible), $A\_1,\; \backslash ,\; satisfied\; A\_2,\; \backslash \; dots$:

$P\; (A\_1\; \backslash \; cup\; A\_2\; \backslash \; cup\; \backslash \; cdots)\; =\; \backslash \; sum\_\; \{I\; =\; 1\}\; ^\; \{+\; \backslash \; infty\}\; P\; (A\_i)$.

I.e. the probability of an event which is the meeting (countable) disjoined events is equal to the sum of the probabilities of these events. This is called σ-additivity, or countable additivity (if the events are not two to two not disjoined, this relation is not truer in general).

Consequences

Starting from the axioms, are shown a certain number of useful properties for the probability theory, for example:

• : $P\; (\backslash \; emptyset)\; =0$.

• : if $\backslash \; A$, $\backslash \; B$ is two incompatible events, then $P\; (has\; \backslash \; cup\; B)\; =\; P\; (A)\; +\; P\; (B)$.

• : for all events $\backslash \; A$, $\backslash \; B$, $P\; (\backslash \; cup\; B)\; =\; P\; (A)\; has\; +\; P\; (B)\; -\; P\; (has\; \backslash \; course\; B)$.

This means that the probability so that one at least events $\backslash \; A$ or $\backslash \; B$ is carried out is equal to the sum of the probabilities so that $\backslash \; A$ is carried out, and so that $\backslash \; B$ is carried out, less probability so that $\backslash \; A$ and $\backslash \; B$ is carried out simultaneously.

• : for any event $\backslash \; A$, $P\; (\backslash \; Omega\; \backslash \; setminus\; A)\; =\; 1\; -\; P\; (A)$.

This means that the probability so that event does not occur is equal to 1 minus the probability so that it is carried out; this property is used when it is simpler to determine the probability of the contrary event than that of the event.

• : $P\; (B\; \backslash \; setminus\; A)\; =\; P\; (B)\; -\; P\; (has\; \backslash \; course\; B)$; in particular, if $A\; \backslash \; subset\; B$, then $P\; (B\; \backslash \; setminus\; A)\; =\; P\; (B)\; -\; P\; (A)$

(it results from it that if $A\; \backslash \; subset\; B$, then $P\; (A)\; \backslash \; Leq\; P\; (B)$: it is the property of growth of the probability).

the preceding relation means that the probability that B is carried out, but not has, is equal to the difference $P\; (B)\; -\; P\; (has\; \backslash \; course\; B)$.

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