Disjoint events A and B with positive probability must not be independent as $P(A)$, $P(B) \gt 0$, $P(A\cap B) = P(\emptyset) = 0 \Rightarrow P(A\cap B) \lt P(A)P(B)$. What are some real world examples that can help better understanding this?
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Consider any attribute which is normally regarded as categorical (eye color, sex, educational level). Any of them work - e.g. if I have blue eyes, even though brown eyes are very common, I know I don't have brown eyes - not-having-brown-eyes is utterly dependent on having-blue-eyes.
[If your eyes are brown, they aren't blue.* If your eyes are blue, they aren't brown. If one occurs, the other doesn't.]
* Note that "one blue and one brown eye" is neither "having brown eyes" nor "having blue eyes".