Let (Ω, F, P) be a probability space and let A, B, C be events with P(C) > 0. We call A and B (conditional) independent given C if and only if P(A ∩ B|C) = P(A|C)P(B|C). (a) Show that conditional independence can be understood as independence with respect to a suitably chosen probability measure P˜ on (Ω, F). (b) There are two coins in a box. One coin is a fair coin, the other coin is a fake coin with a head on both sides. We blindly take a coin from the box and toss it. Then we throw it remaining coin. We watch the events A = 'the first coin lands heads'; B = 'the second coin lands heads';
Let (Ω, F, P) be a
B (conditional) independent given C if and only if
P(A ∩ B|C) = P(A|C)P(B|C).
(a) Show that conditional independence can be understood as independence with respect to a suitably chosen probability measure P˜ on (Ω, F).
(b) There are two coins in a box. One coin is a fair coin,
the other coin is a fake coin with a head on both sides. We
blindly take a coin from the box and toss it. Then we throw it
remaining coin. We watch the events
A = 'the first coin lands heads';
B = 'the second coin lands heads';
C = 'the coin that is taken first is the fair coin'.
Determine whether the following statements and explain why they are true or false:
1. A and B are independent.
2. A and B are given independently C.
Provide both intuitive reasoning and formal proof.
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