Probability space 43325 225027620 2008-07-11T15:09:09Z 128.40.42.4 /* Defining the events in terms of the sample space */ A '''probability space''' is an abstract mathematical object involving a collection of states of the world, a collection of events which are [[set]]s into which states can be (not necessarily uniquely) assigned, and a certain [[probability]] given for each event. Its definition as a special case of [[measure space]] is the foundation of [[probability theory]]. It was introduced by [[Andrey Kolmogorov|Kolmogorov]] in the 1930s. For an algebraic alternative to Kolmogorov's approach, see [[algebra of random variables]]. ==Definition== A probability space <math>(\Omega, \mathcal F, P)</math> is a [[measure (mathematics)|measure space]] with a measure ''P'' that satisfies the [[probability axioms]]. The '''sample space''' <math>\Omega,</math> is a nonempty [[set]] whose elements are known as '''outcomes''' or '''states of nature''' and are often given the symbol <math>\omega.</math>The set of all the possible outcomes of an experiment is known as the sample space of the experiment. === Events === The second item, <math>\mathcal F </math>, is a [[sigma-algebra|σ-algebra]] of subsets of <math>\Omega</math>. Its elements are called [[Event (probability theory)|events]], which are sets of outcomes for which one can ask a probability. Because <math>\mathcal F</math> is a σ-algebra, it contains <math>\Omega</math>; also, the complement of any event is an event, and the union of any (finite or countably infinite) sequence of events is an event. Usually, the events are the [[Lebesgue measure|Lebesgue-measurable]] or [[Borel measure|Borel-measurable]] sets of real numbers. === Probability measure === The '''probability measure''' <math>P</math> is a function from <math>\mathcal F</math> to the real numbers that assigns to each event a ''probability'' between 0 and 1. It must satisfy the [[probability axioms]]. Because <math>P</math> is a function defined on <math>\mathcal F</math> and not on <math>\Omega</math>, the set of ''events'' is not required to be the complete [[power set]] of the ''sample space''; that is, not every set of outcomes is necessarily an event. When more than one measure is under discussion, probability measures are often written in [[blackboard bold]] to distinguish them. When there is only one probability measure under discussion, it is often denoted by '''Pr''', meaning "probability of". ==Related concepts== ===Probability distribution=== Any [[probability distribution]] defines a probability measure. ===Random variables=== A [[random variable]] ''X'' is a [[measurable function]] from the ''sample space'' <math>\Omega</math>; to another measurable space called the '''state space'''. If ''X'' is a [[real]]-valued random variable, then the notation <math>{\scriptstyle\Pr(X \geq 60)}</math> is shorthand for <math>{\scriptstyle\Pr(\{ \omega \in \Omega \mid X(\omega) \geq 60 \})}</math>, assuming that <math>{\scriptstyle X \geq 60}</math> is an event. ===Defining the events in terms of the sample space=== If <math>\Omega</math> is [[countable]] we almost always define <math>\mathcal F</math> as the [[power set]] of <math>\Omega</math>, i.e <math>\mathcal F=\mathbb P (\Omega)</math> which is trivially a σ-algebra and the biggest one we can create using <math>\Omega</math>. We can therefore omit <math>\mathcal{F}</math> and just write <math>(\Omega,\ P)</math> to define the probability space. On the other hand, if <math>\Omega</math> is [[uncountable]] and we use <math>\mathcal F=\mathbb P (\Omega)</math> we get into trouble defining our probability measure <math>P</math> because <math>\mathcal{F}</math> is too 'huge', i.e. there will often be sets to which it will be impossible to assign a unique measure, giving rise to problems like the [[Banach–Tarski paradox]]. In this case, we have to use a smaller σ-algebra <math>\mathcal F</math> (e.g. the [[Borel algebra]] of <math>\Omega</math>, which is the smallest σ-algebra that makes all open sets measurable). ===Conditional probability=== Kolmogorov's definition of probability spaces gives rise to the natural concept of [[conditional probability]]. Every set <math>A</math> with non-zero probability (that is, ''P(A) > 0'' ) defines another probability measure : <math>P(B \vert A) = {P(B \cap A) \over P(A)}</math> on the space. This is usually read as the "probability of ''B'' given ''A''". ===Independence=== Two events, ''A'' and ''B'' are said to be [[Statistical independence|independent]] if ''P''(''A''∩''B'')=''P''(''A'')''P''(''B''). Two random variables, ''X'' and ''Y'', are said to be independent if any event defined in terms of ''X'' is independent of any event defined in terms of ''Y''. Formally, they generate independent σ-algebras, where two σ-algebras ''G'' and ''H'', which are subsets of ''F'' are said to be independent if any element of ''G'' is independent of any element of ''H''. The concept of independence is where probability theory departs from [[measure theory]]. ===Mutual exclusivity=== Two events, ''A'' and ''B'' are said to be [[mutually exclusive]] or ''disjoint'' if ''P''(''A''∩''B'')=0. (This is weaker than ''A''∩''B''=<span class="Unicode">∅</span>, which is the definition of [[disjoint]] for sets). If ''A'' and ''B'' are disjoint events, then ''P''(''A''∪''B'')=''P''(''A'')+''P''(''B''). This extends to a (finite or countably infinite) sequence of events. However, the probability of the union of an uncountable set of events is not the sum of their probabilities. For example, if Z is a [[normal distribution|normally distributed]] random variable, then ''P''(''Z''=''x'') is 0 for any ''x'', but ''P''(''Z'' is real)=1. The event ''A''∩''B'' is referred to as ''A AND B'', and the event ''A''∪''B'' as ''A OR B''. ==Examples== ===First example=== If the space concerns one flip of a fair coin, then the outcomes are heads and tails: <math>\Omega = \{H,T\}</math> The events are *{H}: heads, *{T}: tails, *{}: neither heads nor tails, and *{H,T}: heads or tails. So, <math>F=\{\{H\},\{T\},\{\},\{H,T\}\}.</math> There is a fifty percent chance of tossing either heads or tail: P({H}) = P({T}) = 0.5. The chance of tossing neither is zero: P({})=0, and the chance of tossing one or the other is one: P({H,T})=1. ===Second example=== If 100 voters are to be drawn randomly from among all voters in California and asked whom they will vote for governor, then the set of all sequences of 100 Californian votes would be the sample space Ω. The set of all sequences of 100 Californian voters in which at least 60 will vote for Schwarzenegger is identified with the "event" that at least 60 of the 100 chosen voters will so vote. Then, <math> \mathcal F </math> contains: (1) the set of all sequences of 100 where at least 60 vote for [[Schwarzenegger]]; (2) the set of all sequences of 100 where fewer than 60 vote for [[Schwarzenegger]] (the converse of (1)); (3) the sample space Ω as above; and (4) the empty set. An example of a random variable is the number of voters who will vote for Schwarzenegger in the sample of 100. == Bibliography == * Pierre Simon de Laplace (1812) ''Analytical Theory of Probability'' :: The first major treatise blending calculus with probability theory, originally in French: ''Théorie Analytique des Probabilités''. *Andrei Nikolajevich Kolmogorov (1950) ''Foundations of the Theory of Probability'' :: The modern measure-theoretic foundation of probability theory; the original German version (''Grundbegriffe der Wahrscheinlichkeitrechnung'') appeared in 1933. * Harold Jeffreys (1939) ''The Theory of Probability'' :: An empiricist, Bayesian approach to the foundations of probability theory. * Edward Nelson (1987) ''Radically Elementary Probability Theory'' :: Discrete foundations of probability theory, based on nonstandard analysis and internal set theory. downloadable. http://www.math.princeton.edu/~nelson/books.html * Patrick Billingsley: ''Probability and Measure'', John Wiley and Sons, New York, Toronto, London, 1979. * Henk Tijms (2004) ''Understanding Probability '' :: A lively introduction to probability theory for the beginner, Cambridge Univ. 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