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| {{Probability fundamentals}}
| | Hello! My name is Jodie. <br>It is a little about myself: I live in Italy, my city of Malalbergo. <br>It's called often Northern or cultural capital of BO. I've married 2 years ago.<br>I have two children - a son (Verona) and the daughter (Raymundo). We all like Equestrianism.<br>[http://www.eurosummit13.com/content/forevertranscriptioncomlegal-transcription-servicesphp-blank-techniques legal transcription courses] |
| In Kolmogorov's [[probability theory]], the [[probability]] ''P'' of some [[event (probability theory)|event]] ''E'', denoted <math>P(E)</math>, is usually defined in such a way that ''P'' satisfies the '''Kolmogorov axioms''', named after the famous [[Russia|Russian]] [[mathematician]] [[Andrey Kolmogorov]], which are described below.
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| 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''.
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| An alternative approach to formalising probability, favoured by some [[Bayesian theory|Bayesians]], is given by [[Cox's theorem]].
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| == Axioms ==
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| === First axiom ===
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| The probability of an event is a non-negative real number:
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| :<math>P(E)\in\mathbb{R}, P(E)\geq 0 \qquad \forall E\in F</math>
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| where <math>F</math> is the event space. In particular, <math>P(E)</math> is always finite, in contrast with more general [[measure theory]]. Theories which assign [[negative probability]] relax the first axiom.
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| === Second axiom ===
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| {{seealso|Unitarity (physics)}}
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| 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.
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| : <math>P(\Omega) = 1.</math>
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| This is often overlooked in some mistaken probability calculations; if you cannot precisely define the whole sample space, then the probability of any subset cannot be defined either.
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| ===Third axiom===
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| This is the assumption of [[σ-additivity]]:
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| : Any [[countable]] sequence of [[disjoint sets|disjoint]] (synonymous with ''mutually exclusive'') events <math>E_1, E_2, ...</math> satisfies
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| ::<math>P(E_1 \cup E_2 \cup \cdots) = \sum_{i=1}^\infty P(E_i).</math>
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| Some authors consider merely [[finitely additive]] probability spaces, in which case one just needs an [[algebra of sets]], rather than a [[σ-algebra]]. [[Quasiprobability distribution]]s in general relax the third axiom.
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| ==Consequences==
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| From the [[Kolmogorov]] axioms, one can deduce other useful rules for calculating probabilities.
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| ===Monotonicity===
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| : <math>\quad\text{if}\quad A\subseteq B\quad\text{then}\quad P(A)\leq P(B).</math>
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| ===The probability of the empty set===
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| : <math>P(\emptyset)=0.</math>
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| ===The numeric bound===
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| It immediately follows from the monotonicity property that
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| : <math>0\leq P(E)\leq 1\qquad \text{∀} E\in F.</math>
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| ==Proofs==
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| The proofs of these properties are both interesting and insightful. They illustrate the power of the third axiom,
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| and its interaction with the remaining two axioms. When studying [[axiomatic]] [[probability theory]], many deep consequences follow from merely these three axioms.
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| In order to verify the monotonicity property, we set <math>E_1=A</math> and <math>E_2=B\backslash A</math>,
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| where <math>\quad A\subseteq B \text{ and } E_i=\emptyset</math> for <math>i\geq 3</math>. It is easy to see that the sets <math>E_i</math>
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| are pairwise disjoint and <math>E_1\cup E_2\cup\ldots=B</math>. Hence,
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| we obtain from the third axiom that
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| : <math>P(A)+P(B\backslash A)+\sum_{i=3}^\infty P(\emptyset)=P(B).</math>
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| Since the left-hand side of this equation is a series of non-negative numbers, and that it converges to
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| <math>P(B)</math> which is finite, we obtain both <math>P(A)\leq P(B)</math> and <math>P(\emptyset)=0</math>.
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| The second part of the statement is seen by contradiction: if <math>P(\emptyset)=a</math> then the left hand side is not less than
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| : <math>\sum_{i=3}^\infty P(E_i)=\sum_{i=3}^\infty P(\emptyset)=\sum_{i=3}^\infty a = \begin{cases} 0 & \text{if } a=0, \\ \infty & \text{if } a>0. \end{cases}</math>
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| If <math>a>0</math> then we obtain a contradiction, because the sum does not exceed <math>P(B)</math> which is finite. Thus, <math>a=0</math>. We have shown as a byproduct of the proof of monotonicity that <math>P(\emptyset)=0</math>.
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| ==More consequences==
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| Another important property is:
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| : <math>P(A \cup B) = P(A) + P(B) - P(A \cap B).</math>
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| This is called the addition law of probability, or the sum rule.
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| That is, the probability that ''A'' ''or'' ''B'' will happen is the sum of the
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| probabilities that ''A'' will happen and that ''B'' will happen, minus the
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| probability that both ''A'' ''and'' ''B'' will happen. This can be extended to the [[inclusion-exclusion principle]].
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| : <math>P(\Omega\setminus E) = 1 - P(E)</math>
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| That is, the probability that any event will ''not'' happen is 1 minus the probability that it will.
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| ==Simple example: coin toss==
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| Consider a single coin-toss, and assume that the coin will either land heads (H) or tails (T) (but not both). No assumption is made as to whether the coin is fair.
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| We may define: | |
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| : <math>\Omega = \{H,T\}</math>
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| : <math>F = \{\emptyset, \{H\}, \{T\}, \{H,T\}\}</math>
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| Kolmogorov's axioms imply that:
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| : <math>P(\emptyset) = 0</math>
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| The probability of ''neither'' heads ''nor'' tails, is 0.
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| : <math>P(\{H,T\}) = 1</math>
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| The probability of ''either'' heads ''or'' tails, is 1.
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| : <math>P(\{H\}) + P(\{T\}) = 1</math>
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| The sum of the probability of heads and the probability of tails, is 1.
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| == See also ==
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| * [[Law of total probability]]
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| * [[Measure (mathematics)]]
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| * [[Borel Algebra]]
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| * [[Sigma-algebra | σ-Algebra]]
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| * [[Probability theory]]
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| * [[Set theory]]
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| * [[Conditional probability]]
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| * [[Quasiprobability]]
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| {{No footnotes|date=November 2010}}
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| ==Further reading==
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| * Von Plato, Jan, 2005, "Grundbegriffe der Wahrscheinlichkeitsrechnung" in [[Ivor Grattan-Guinness|Grattan-Guinness, I.]], ed., ''Landmark Writings in Western Mathematics''. Elsevier: 960-69. (in English)
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| * {{citation|author1=Glenn Shafer|author2=Vladimir Vovk|title=The origins and legacy of Kolmogorov’s Grundbegriffe|url=http://www.probabilityandfinance.com/articles/04.pdf}}
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| ==External links==
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| * [http://plato.stanford.edu/entries/probability-interpret/#KolProCal Kolmogorov`s probability calculus], Stanford Encyclopedia of Philosophy.
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| * [http://mws.cs.ru.nl/mwiki/prob_1.html#M2 Formal definition] of probability in the [[Mizar system]], and the [http://mmlquery.mizar.org/cgi-bin/mmlquery/emacs_search?input=(symbol+Probability+%7C+notation+%7C+constructor+%7C+occur+%7C+th)+ordered+by+number+of+ref list of theorems] formally proved about it.
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| {{DEFAULTSORT:Probability Axioms}}
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| [[Category:Probability theory]]
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| [[Category:Mathematical axioms]]
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Hello! My name is Jodie.
It is a little about myself: I live in Italy, my city of Malalbergo.
It's called often Northern or cultural capital of BO. I've married 2 years ago.
I have two children - a son (Verona) and the daughter (Raymundo). We all like Equestrianism.
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