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Law of total probability

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In statistics, the law of total probability is that "the prior probability of A is equal to the prior expected value of the posterior probability of A." That is, for any random variable N,

\Pr(A)=E(\Pr(A\mid N))

where \Pr(A\mid N) is the conditional probability of A given N.

Law of alternativesEdit

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 variables. It is the proposition that if { Bn : n = 1, 2, 3, ... } is a finite or countably infinite partition of a probability space and each set Bn is measurable, then for any event A we have

\Pr(A)=\sum_{n} \Pr(A\cap B_n)\,

or, alternatively,

\Pr(A)=\sum_{n} \Pr(A\mid B_n)\Pr(B_n).\,

See alsoEdit

de:Totale Wahrscheinlichkeit]]
it:Teorema della probabilità totale]]
zh:全概率公式]]
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