Grimmett and Stirzaker 1.5.5.b

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.

Thanks

• Knowing that the smaller die is $3$ is evidence that the larger die came in above average (as we know it couldn't have been a $1$ or a $2$). – lulu Jun 26 '18 at 11:02
• Note: given that there are only $36$ possible rolls, I'd say a perfectly good method is to simply list them all and compute everything by hand. Having done that, you might search for a more conceptual approach but at least you will be certain of the answers. – lulu Jun 26 '18 at 11:05