Is the Law of Total Probability a Theorem or an Axiom?

[TSC]

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

 

[homeomorphic]

Theorem.

 

[jbunniii]

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.

 

[TSC]

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