Probability axioms

Probabiliti  tina sababaraha kajadian  (dilambangkeun ku ) is defined with respect to a "universe" or sample space  of all possible elementary events in such a way that  must satisfy the Kolmogorov axioms.

Alternatively, a probability can be interpreted as a measure on a σ-algebra of subsets of the sample space, those subsets being the events, such that the méasure of the whole set equals 1.  This property is important, since it gives rise to the natural concept of conditional probability.  Every set  with non-zero probability defines another probability

 

on the space.  This is usually réad as "probability of  given ".  If the conditional probability of  given  is the same as the probability of , then  and  are said to be independent.

In the case that the sample space is finite or countably infinite, a probability function can also be defined by its values on the elementary events  where .

Kolmogorov axioms 

The following three axioms are known as the Kolmogorov axioms, after Andrey Kolmogorov who developed them.

First axiom 

 For any set , .

That is, the probability of an event set is represented by a réal number between 0 and 1.

Second axiom 

 

That is, the probability that some elementary event in the entire sample set will occur is 1.  More specifically, there are no elementary events outside the sample set.

This is often overlooked in some mistaken probability calculations; if you cannot precisely define the whole sample set, then the probability of any subset cannot be defined either.

Third axiom 

 Any countable sequence of mutually disjoint events  satisfies .

That is, the probability of an event set which is the union of other disjoint subsets is the sum of the probabilities of those subsets.  This is called σ-additivity.  If there is any overlap among the subsets this relation does not hold.

For an algebraic alternative to Kolmogorov's approach, see algebra of random variables.

Lemmas in probability 

From the Kolmogorov axioms one can deduce other useful rules for calculating probabilities:

 

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 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.

Using conditional probability as defined above, it also follows immediately that

 

That is, the probability that A and B will happen is the probability
that A will happen, times the probability that B will happen given
that A happened; this relationship gives Bayes' theorem. It then follows that A and B are independent if and only if

 .

See also 

 frequency probability
 personal probability
 pilihan probabiliti
 statistical regularity

Probability theory