Probability space

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, a probability space or a probability triple 
 
 
 
 (
 Ω
 ,
 
 
 F
 
 
 ,
 P
 )
 
 
 {\displaystyle (\Omega ,{\mathcal {F}},P)}
 
 is a <a href="/facts/Space_(mathematics)/G6rt4bQ8">mathematical construct</a> that provides a formal model of a <a href="/facts/Randomness/QJQBoEX5">random</a> process or "experiment". For example, one can define a probability space which models the throwing of a <a href="/facts/Dice/qBnylzwJ">die</a>.
A probability space consists of three elements:

<ol><li>A <a href="/facts/Sample_space/WVldkN4F">sample space</a>, 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
, which is the set of all possible <a href="/facts/Outcome_(probability)/1hdJcoQR">outcomes</a> of a random process under consideration.</li>
<li>An event space, 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
, which is a set of <a href="/facts/Event_(probability_theory)/aJ0l4oQb">events</a>, where an event is a subset of outcomes in the sample space.</li>
<li>A <a href="/facts/Probability_measure/8BjhgoPR">probability function</a>, 
 
 
 
 P
 
 
 {\displaystyle P}
 
, which assigns, to each event in the event space, a <a href="/facts/Probability/fgYvMhwb">probability</a>, which is a number between 0 and 1 (inclusive).</li></ol>
In order to provide a model of probability, these elements must satisfy <a href="/facts/Probability_axioms/j6IkXsIT">probability axioms</a>.
In the example of the throw of a standard die,

<ol><li>The sample space 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 is typically the set 
 
 
 
 {
 1
 ,
 2
 ,
 3
 ,
 4
 ,
 5
 ,
 6
 }
 
 
 {\displaystyle \{1,2,3,4,5,6\}}
 
 where each element in the set is a label which represents the outcome of the die landing on that label. For example, 
 
 
 
 1
 
 
 {\displaystyle 1}
 
 represents the outcome that the die lands on 1.</li>
<li>The event space 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 could be the <a href="/facts/Power_set/FVuf8PDD">set of all subsets</a> of the sample space, which would then contain simple events such as 
 
 
 
 {
 5
 }
 
 
 {\displaystyle \{5\}}
 
 ("the die lands on 5"), as well as complex events such as 
 
 
 
 {
 2
 ,
 4
 ,
 6
 }
 
 
 {\displaystyle \{2,4,6\}}
 
 ("the die lands on an even number").</li>
<li>The probability function 
 
 
 
 P
 
 
 {\displaystyle P}
 
 would then map each event to the number of outcomes in that event divided by 6 – so for example, 
 
 
 
 {
 5
 }
 
 
 {\displaystyle \{5\}}
 
 would be mapped to 
 
 
 
 1
 
 /
 
 6
 
 
 {\displaystyle 1/6}
 
, and 
 
 
 
 {
 2
 ,
 4
 ,
 6
 }
 
 
 {\displaystyle \{2,4,6\}}
 
 would be mapped to 
 
 
 
 3
 
 /
 
 6
 =
 1
 
 /
 
 2
 
 
 {\displaystyle 3/6=1/2}
 
.</li></ol>
When an experiment is conducted, it results in exactly one outcome 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 from the sample space 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
. All the events in the event space 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 that contain the selected outcome 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 are said to "have occurred". The probability function 
 
 
 
 P
 
 
 {\displaystyle P}
 
 must be so defined that if the experiment were repeated arbitrarily many times, the number of occurrences of each event as a fraction of the total number of experiments, will most likely tend towards the probability assigned to that event.
The Soviet mathematician <a href="/facts/Andrey_Kolmogorov/bEKwtrs6">Andrey Kolmogorov</a> introduced the notion of a probability space and the <a href="/facts/Axioms_of_probability/j6IkXsIT">axioms of probability</a> in the 1930s. In modern probability theory, there are alternative approaches for axiomatization, such as the <a href="/facts/Algebra_of_random_variables/CD0E9PYT">algebra of random variables</a>.

Probability space open-in-new

Probability space