1. Let A, B and C be three events. (a) Prove or disprove each of the following statements : (1)(5 points) A and B are c

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1. Let A, B and C be three events.
(a) Prove or disprove each of the following statements :
(1)(5 points) A and B are conditionally independent given C if
A, B, C are independent.
(2) (5 points) Suppose that
• A and B are conditionally independent given C, and
• A and B are conditionally independent given C c .
Then, A and B are not necessarily independent but A and B are
independent if C is independent of either A or B.
(3) (5 points) A necessary condition of P(A|B) > P(A|Bc ),
P(A|B ∩ C) < P(A|Bc∩C) and P(A|B∩C c ) < P(A|Bc∩C c ) is
either 1 P(C|B) > P(C|Bc ) and P(A|B ∩ C) > P(A|B ∩ C c ), or
alternatively 2 P(C|B) < P(C|Bc ), P(A|B ∩ C) < P(A|B ∩ C c
).
(b) (5 points) Provide an example of Simpson’s paradox in real
life (except for examples discussed in the lecture notes) with a
description of three events A,B,C and corresponding conditional
probabilities in detail