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.

 

[blue_raver22]

The Law of Total Probability is a theorem, not an axiom. Axioms are basic assumptions or principles that are accepted without proof, while theorems are statements that can be proven based on axioms and other previously proven theorems. The Law of Total Probability can be proven using basic axioms of probability and other theorems, such as the rules of conditional probability and the multiplication rule. Therefore, it is considered a theorem rather than an axiom.