# Law of total probability

### From Bvio.com

Nomenclature in probability theory is not wholly standard.

*Sometimes* the phrase * law of total probability* refers to the proposition that if {

*B*

_{n}:

*n*= 1, 2, 3, ... } is a finite or countably infinite partition of a probability space and each set

*B*

_{n}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*(*B*_{n}). 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.