Everything about Kolmogorov Axioms totally explained
In
probability theory, the
probability P of some
event E, denoted
, is defined in such a way that
P satisfies the
Kolmogorov axioms, named after
Andrey Kolmogorov.
These assumptions can be summarised as: Let (Ω,
F,
P) be a
measure space with
P(Ω)=1. Then (Ω,
F,
P) is a
probability space, with sample space Ω, event space
F and probability measure
P.
First axiom
The probability of an event is a non-negative real number:
»
where
is the event space.
Second axiom
This is the assumption of
unit measure: that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space.
»
This is often overlooked in some mistaken probability calculations; if you can't precisely define the whole sample space, then the probability of any subset can't be defined either.
Third axiom
This is the assumption of
σ-additivity:
» Any
countable sequence of pairwise disjoint events
satisfies
Some authors consider merely
finitely-additive probability spaces, in which case one just needs an
algebra of sets, rather than a
σ-algebra.
Consequences
From the Kolmogorov axioms one can deduce other useful rules for calculating probabilities:
»
This is called the addition law of probability, or the sum rule.
That is, the probability that
A or B will happen is the sum of the
probabilities that
A will happen and that
B will happen, minus the
probability that both
A and B will happen. This can be extended to the
inclusion-exclusion principle.
»
That is, the probability that any event will
not happen is 1 minus the probability that it will.
Further Information
Get more info on 'Kolmogorov Axioms'.
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