# Law of total probability

Is the law of total probability a theorem or an axiom?

## Answers and Replies

Related Set Theory, Logic, Probability, Statistics News on Phys.org
Theorem.

jbunniii
Science Advisor
Homework Helper
Gold Member
It is an axiom that the probabilities of disjoint events can be summed: if ##A_1, \ldots A_N## are disjoint and ##\bigcup_{n=1}^{N}A_n = A##, then ##P(A) = \sum_{n=1}^{N} P(A_n)##.

If ##B \subset A##, then we may write ##B## as the disjoint union ##B = \bigcup_{n=1}^{N} (B \cap A_n)##, so the axiom gives us ##P(B) = \sum_{n=1}^{N}P(B \cap A_n)##.

Finally, if ##P(A_n) > 0## we define ##P(B | A_n) = P(B \cap A_n) / P(A_n)##, so ##P(B \cap A_n) = P(B|A_n) P(A_n)##. Substituting into the result in the previous paragraph, we obtain
$$P(B) = \sum_{n=1}^{N} P(B|A_n) P(A_n)$$

So, it's a theorem, but quite a simple one: we simply substitute a definition into an axiom.

wow!
That is really very clear. :) Thanks.