The question asks to show an example to show that conditional independence does not imply (nor is it implied by) independence.
In the solutions they include Example b:
Roll two dice; let A be the event that the smaller is 3, let B be the event that the larger is 6, and let C be the event that the smaller score is no more than 3, and the larger is 4 or more. Then A and B are conditionally independent given C, but not independent.
1.) Why are A and B not independent? They correspond to different dice and the probabilities are P(A) = 1/3 and P(B) = 1/6. The probability of $P(A \cap B)$ is 2/36 which is 1/18 and therefore equal to P(A)p(B)?
2.) I'm also struggling to see how C would make the conditionally independent, but maybe that will be clearer when I have an answer to the first question.