Prove or disprove via proof that events $X$ and $Y$ can be independent and mutually exclusive if both of their probabilities are greater than $0$.
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Two events $X$ and $Y$ are independent if and only if $P(X \cap Y) = P(X)P(Y)$. They are mutually exclusive if and only if $P(X \cap Y) = 0$.
In this case, $P(X) > 0$ and $P(Y) > 0$, which means that $P(X)P(Y) > 0$. If the events are independent, then $P(X)P(Y) = P(X \cap Y) > 0$, which would imply that they are not mutually exclusive!
In other words, the events cannot be both independent and mutually exclusive if both of their probabilities are greater than $0$.
