So far I have:
P(AnB)=P(a)*P(B) because the two events are independent (given in statement). I know we need to show:
P(AnB|C)=P(A|C)*P(B|C).. Is this right?
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You might try $C = (A \cap B)^c$. – Robert Israel Oct 01 '18 at 00:17
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So is the answer that if the two events are independent then they are not independent conditional to C – Taffies1 Oct 01 '18 at 00:32
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No, they may or may not be conditionally independent. – Robert Israel Oct 01 '18 at 01:07
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Im confused by what you mean by "may or may not be". I am trying to prove: – Taffies1 Oct 01 '18 at 01:08
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Is there any way to prove that this statement is mathematically true or do I assume it is just false? – Taffies1 Oct 01 '18 at 01:09
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There are some $C$ such that $P(A \cap B \mid C) = P(A \mid C) P(B \mid C)$. There may be some $C$ such that $P(A \cap B \mid C) \ne P(A \mid C) P(B \mid C)$. – Robert Israel Oct 01 '18 at 01:11
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And thus the statement is false. – Robert Israel Oct 01 '18 at 01:12
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Can you give an example of which A and B are independent but not independent conditional to C? – Taffies1 Oct 01 '18 at 01:23
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@Taffies1 Such an example was already given in the question that we marked this a duplicate of. – JMoravitz Oct 01 '18 at 16:24