Editing Law of total probability

{{Short description|Concept in probability theory}}
{{Probability fundamentals}}

In [[probability theory]], the '''law''' (or '''formula''') '''of total probability''' is a fundamental rule relating [[Marginal probability|marginal probabilities]] to [[conditional probabilities]]. It expresses the total probability of an outcome which can be realized via several distinct [[Event (probability theory)|events]], hence the name.

==Statement==
The law of total probability is<ref name= ZK>Zwillinger, D., Kokoska, S. (2000) ''CRC Standard Probability and Statistics Tables and Formulae'', CRC Press. {{isbn|1-58488-059-7}} page 31.</ref> a [[theorem]] that states, in its discrete case, if <math>\left\{{B_n : n = 1, 2, 3, \ldots}\right\}</math> is a finite or [[Countable set|countably infinite]] set of [[mutually exclusive]] and [[collectively exhaustive]] events, then for any event <math>A</math>

:<math>P(A)=\sum_n P(A\cap B_n)</math>

or, alternatively,<ref name=ZK/>

:<math>P(A)=\sum_n P(A\mid B_n)P(B_n),</math>

where, for any <math>n</math>, if <math>P(B_n) = 0 </math>, then these terms are simply omitted from the summation since <math>P(A\mid B_n)</math> is finite.

The summation can be interpreted as a [[weighted average]], and consequently the marginal probability, <math>P(A)</math>, is sometimes called "average probability";<ref name="Pfeiffer1978">{{cite book|author=Paul E. Pfeiffer|title=Concepts of probability theory|url=https://books.google.com/books?id=_mayRBczVRwC&pg=PA47|year=1978|publisher=Courier Dover Publications|isbn=978-0-486-63677-1|pages=47–48}}</ref> "overall probability" is sometimes used in less formal writings.<ref name="Rumsey2006">{{cite book|author=Deborah Rumsey|authorlink= Deborah J. Rumsey |title=Probability for dummies|url=https://books.google.com/books?id=Vj3NZ59ZcnoC&pg=PA58|year=2006|publisher=For Dummies|isbn=978-0-471-75141-0|page=58}}</ref>

The law of total probability can also be stated for conditional probabilities: 

:<math>P( {A|C} ) = \frac{{P( {A,C} )}}{{P( C )}} = \frac{{\sum\limits_n {P( {A,{B_n},C} )} }}{{P( C )}} = \frac{{\sum\limits_n P ( {A\mid {B_n},C} )P( {{B_n}\mid C} )P( C )}}{{P( C )}} = \sum\limits_n P ( {A\mid {B_n},C} )P( {{B_n}\mid C} )</math>

Taking the <math>B_n</math> as above, and assuming <math>C</math> is an event [[Independence (probability theory)|independent]] of any of the <math>B_n</math>:

:<math>P(A \mid C) = \sum_n P(A \mid C,B_n) P(B_n) </math>

==Continuous case==
The law of total probability extends to the case of conditioning on events generated by continuous random variables. Let <math>(\Omega, \mathcal{F}, P) </math> be a [[probability space]]. Suppose <math> X </math> is a random variable with distribution function <math>F_X</math>, and <math>A</math> an event on <math>(\Omega, \mathcal{F}, P) </math>. Then the law of total probability states 

<math>P(A) = \int_{-\infty}^\infty P(A |X = x) d F_X(x). </math>

If <math>X</math> admits a density function <math>f_X</math>, then the result is 

<math>P(A) = \int_{-\infty}^\infty P(A |X = x) f_X(x) dx. </math>

Moreover, for the specific case where <math>A = \{Y \in B \}</math>, where <math>B</math> is a Borel set, then this yields 

<math>P(Y \in B) = \int_{-\infty}^\infty P(Y \in B |X = x) f_X(x) dx. </math>

==Example==
Suppose that two factories supply [[light bulb]]s to the market. Factory ''X''<nowiki>'</nowiki>s bulbs work for over 5000 hours in 99% of cases, whereas factory ''Y''<nowiki>'</nowiki>s bulbs work for over 5000 hours in 95% of cases. It is known that factory ''X'' supplies 60% of the total bulbs available and Y supplies 40% of the total bulbs available. What is the chance that a purchased bulb will work for longer than 5000 hours?

Applying the law of total probability, we have:

: <math>
\begin{align}
P(A) & = P(A\mid B_X) \cdot P(B_X) + P(A\mid B_Y) \cdot P(B_Y) \\[4pt]
& = {99 \over 100} \cdot {6 \over 10} + {95 \over 100} \cdot {4 \over 10} = {{594 + 380} \over 1000} = {974 \over 1000}
\end{align}
</math>

where
* <math>P(B_X)={6 \over 10}</math> is the probability that the purchased bulb was manufactured by factory ''X'';
* <math>P(B_Y)={4 \over 10}</math> is the probability that the purchased bulb was manufactured by factory ''Y'';
* <math>P(A\mid B_X)={99 \over 100}</math> is the probability that a bulb manufactured by ''X'' will work for over 5000 hours;
* <math>P(A\mid B_Y)={95 \over 100}</math> is the probability that a bulb manufactured by ''Y'' will work for over 5000 hours.

Thus each purchased light bulb has a 97.4% chance to work for more than 5000 hours.

==Other names==
The term '''''law of total probability''''' is sometimes taken to mean the '''law of alternatives''', which is a special case of the law of total probability applying to [[discrete random variable]]s.{{Citation needed|date=September 2010}} One author uses the terminology of the "Rule of Average Conditional Probabilities",<ref name="Pitman1993">{{cite book|author=Jim Pitman|title=Probability|url=https://books.google.com/books?id=AoDkBwAAQBAJ&q=pitman%20probability&pg=PA41|year=1993|publisher=Springer|isbn=0-387-97974-3|page=41}}</ref> while another refers to it as the "continuous law of alternatives" in the continuous case.<ref name="Baclawski2008">{{cite book|author=Kenneth Baclawski|title=Introduction to probability with R|url=https://books.google.com/books?id=Kglc9g5IPf4C&pg=PA179|year=2008|publisher=CRC Press|isbn=978-1-4200-6521-3|page=179}}</ref> This result is given by Grimmett and Welsh<ref>''Probability: An Introduction'', by [[Geoffrey Grimmett]] and [[Dominic Welsh]], Oxford Science Publications, 1986, Theorem 1B.</ref>  as the '''partition theorem''', a name that they also give to the related [[law of total expectation]].

==See also==
* [[Law of large numbers]]
* [[Law of total expectation]]
* [[Law of total variance]]
* [[Law of total covariance]]
* [[Law of total cumulance]]
* [[Marginal distribution]]

== Notes ==
<references/>

== References ==
* ''Introduction to Probability and Statistics'' by Robert J. Beaver, Barbara M. Beaver, Thomson Brooks/Cole, 2005, page 159.
* ''Theory of Statistics'', by Mark J. Schervish, Springer, 1995.
* ''Schaum's Outline of Probability, Second Edition'', by John J. Schiller, Seymour Lipschutz, McGraw–Hill Professional, 2010, page 89.
* ''A First Course in Stochastic Models'', by H. C. Tijms, John Wiley and Sons, 2003, pages 431–432.
* ''An Intermediate Course in Probability'', by Alan Gut, Springer, 1995, pages 5–6.

{{DEFAULTSORT:Law Of Total Probability}}
[[Category:Theorems in probability theory]]
[[Category:Statistical laws]]