Probability axioms
From Psychology Wiki
Community portal · Tasks to do · News · Help
Clinical · Educational · Ind&Org · Other fields · Professional · Transpersonal · World
Assessment |
Biopsychology |
Comparative |
Cognitive |
Developmental |
Language
Personality |
Philosophy |
Research Methods |
Social |
Statistics
Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory
The probability P of some event E, denoted 
, is defined with respect to a "universe", or sample space 
, of all possible elementary events in such a way that P 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 measure 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 (that is, P(A)> 0 ) defines another probability
on the space. This is usually read as "probability of B given A". If the conditional probability of B given A is the same as the probability of B, then A and B 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 
Contents |
[edit] Kolmogorov axioms
The following three axioms are known as the Kolmogorov axioms, after Andrey Kolmogorov who developed them. We have an underlying set Ω, a sigma-algebra F of subsets of Ω, and a function P assigning real numbers to members of F. The members of F are those subsets of Ω that are called "events".
[edit] First axiom
- For any set
, that is, for any event 
, we have 
.
That is, the probability of an event is a non-negative real number.
[edit] 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.
[edit] Third axiom
- Any countable sequence of pairwise 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. Some authors consider merely finitely-additive probability spaces, in which case one just needs an algebra of sets, rather than a σ-algebra.
For an algebraic alternative to Kolmogorov's approach, see algebra of random variables.
[edit] Lemmas in probability
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 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
.
[edit] See also
[edit] External links
- The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A.N. Kolmogorov. A.N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A.N. Kolmogorov.
- de:Kolmogorow-Axiom
es:Axiomas de probabilidad fr:Axiomes des probabilités gl:Axiomas de probabilidade he:אקסיומות ההסתברות nl:Axioma's van de kansrekeningro:Axiomele probabilităţii ru:Аксиоматика Колмогорова th:สัจพจน์ของความน่าจะเป็น zh:概率公理
| This page uses content from the English-language version of Wikipedia. The original article was at Probability axioms. The list of authors can be seen in the page history. As with Psychology Wiki, the text of Wikipedia is available under the GNU Free Documentation License. |
