Question

Example of two events that are conditionally independent but not independent

Answer 1

Independent but conditionally dependent

Let's say you flip two fair coins

A - Your first coin flip is heads

B - Your second coin flip is heads

C - Your first two flips were the same

A and B here are independent. However, A and B are conditionally dependent given C, since if you know C then your first coin flip will inform the other one.

Dependent but conditionally independent

I give you a random coin, it can either be biased (favor heads) or fair.

A - Your first coin flip is heads

B - Your second coin flip is heads

C - I gave you a biased coin

A and B here are dependent, since flipping it once will give you evidence on whether or not the coin is biased. This evidence will help inform your next flip.

However the flips are conditionally independent given C.