Conditional Independence I [2 P]

a)

[1 P] Give an example for a probability distribution $P(A,B,C)$ that disproves

$$P(A,B) = P(A) P(B) \rightarrow P(A,B\vert C) = P(A\vert C) P(B\vert C) .$$

Illustrate the phenomenon of 'explaining away'.

b)

[1 P] Give an example for a probability distributions $P(A,B,C)$ that disproves

$$P(A,B\vert C) = P(A\vert C) P(B\vert C) \rightarrow P(A,B) = P(A) P(B).$$