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 { 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
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(
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.