# Law of total probability

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

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

which may be rephrased as

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*.