- #1
mrkb80
- 41
- 0
Hi. I need some help with a proof.
The question says:
Let P be a probability function. Prove that for any finite collection of sets, the sequence A1,A2,...,An of pairwise disjoint sets, P(Union from i=1 to n of Ai)=Ʃ from i=1 to n of Ai
I think there must a mistake in question. My guess is it is supposed to say P(Union from i=1 to n of Ai)=Ʃ from i=1 to n of P(Ai)?
There is a hint with the question:
If A1,A2,...,An are pairwise disjoints on the sample space Ω, then P(Union from i=1 to ∞ Ai) = Ʃ from i=1 to ∞ Ai
If A and B are disjoints defined on Ω then P(A U B) = P(A)+ P(B)
Again, I think there must be a mistake in the hint...P(Union from i=1 to ∞ Ai) = Ʃ from i=1 to ∞ P(Ai)?
My question is are there mistakes and can someone get me started in the right direction (Assuming the question has a mistake, then the book says the proof is a straight forward induction argument with P(A U B) = P(A) + P(B), if A and B are mutually exclusive events over S being the starting point)
The question says:
Let P be a probability function. Prove that for any finite collection of sets, the sequence A1,A2,...,An of pairwise disjoint sets, P(Union from i=1 to n of Ai)=Ʃ from i=1 to n of Ai
I think there must a mistake in question. My guess is it is supposed to say P(Union from i=1 to n of Ai)=Ʃ from i=1 to n of P(Ai)?
There is a hint with the question:
If A1,A2,...,An are pairwise disjoints on the sample space Ω, then P(Union from i=1 to ∞ Ai) = Ʃ from i=1 to ∞ Ai
If A and B are disjoints defined on Ω then P(A U B) = P(A)+ P(B)
Again, I think there must be a mistake in the hint...P(Union from i=1 to ∞ Ai) = Ʃ from i=1 to ∞ P(Ai)?
My question is are there mistakes and can someone get me started in the right direction (Assuming the question has a mistake, then the book says the proof is a straight forward induction argument with P(A U B) = P(A) + P(B), if A and B are mutually exclusive events over S being the starting point)