Independence and conditional independence

When considering independence and conditional independence, there are several well-known examples to show that neither implies the other. However, even though one can intuitively grasp that conditional independence does not imply independence tout-court, there is one special case that challenges intuition a little bit.

We want to show here that if two events A and B are independent conditional to a third event C and also to its complement C^C, then the two events are not necessarily independent, as it is shown in the following example.

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