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 
and 
are independent conditional to a third event 
and also to its complement 
, then the two events are not necessarily independent, as it is shown in the following example.
