Editing Law of total probability (section)

==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>