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Question: (a) Is it possible to have events A, B, C such that P(A|C) < P(B|C) and P(A|Cc) < P(B|Cc), yet P(A) > P(B)? That is, A is less likely than B given that C is true, and also less likely than B given that C is false, yet A is more likely than B if we're given no information about C. Show this is impossible (with a short proof) or find a counterexample (with a story interpreting A, B, C).
(b) If the scenario in (a) is possible, is it a special case of Simpson's paradox, equivalent to Simpson's paradox, or neither? If it is impossible, explain intuitively why it is impossible even though Simpson's paradox is possible.
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