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.


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    $\begingroup$ 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 $2$). $\endgroup$ – lulu Jun 26 '18 at 11:02
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    $\begingroup$ 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. $\endgroup$ – lulu Jun 26 '18 at 11:05

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