Law of total probability

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

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jbunniii
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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.
 
  • #4
TSC
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wow!
That is really very clear. :) Thanks.
 

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