# 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,

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