Law of total probability

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Nomenclature in probability theory is not wholly standard.

Sometimes the phrase law of total probability refers to 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

<math>P(A)=\sum_{n} P(A\cap B_n).</math>

This is also sometimes called the law of alternatives.

The phrase law of total probability is also used to refer to the proposition that says that under similar assumptions, we have

<math>P(A)=\sum_{n} P(A\mid B_n)P(B_n),</math>

which may be rephrased as

<math>P(A)=E(P(A\mid N))</math>

where N is a random variable equal to n with probability P(Bn). It may be stated even more efficiently thus:

The prior probability of A is equal to the prior expected value of the posterior probability of A.

See also law of total expectation, law of total variance.

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