statistics
posted by Mischa .
What is an example of two events that are neither disjoint nor independent?
(I have no idea how that would even work)

throwing 2 dice and flipping a coin.
suppose you want the prob(heads and a sum of 7)
the process of flipping the coin has no effect on the act of throwing the dice.
Thus the events are "independent" 
sorry I misread your question

If the sets are not disjoint then they have elements in common.
for example
the set A containing numbers 1 through 7. There is only one of each number in the set.I can only pick from set A
and
the set B containing numbers 6 through 9. There is only one of each number in the set. You can only pick from set B
Is the probability of picking a number from set B changed by the previous random selection of a number from box A?
Sure it is. If I took the six or the seven from set A, I took it from the intersection, and you can not pick it. Therefore the events are not independent.
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