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.

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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Makenna Lin
Two events $X$ and $Y$ are independent if and only if $P\left(X\cap Y\right)=P\left(X\right)P\left(Y\right)$. They are mutually exclusive if and only if $P\left(X\cap Y\right)=0$.
In this case, $P\left(X\right)>0$ and $P\left(Y\right)>0$, which means that $P\left(X\right)P\left(Y\right)>0$. If the events are independent, then $P\left(X\right)P\left(Y\right)=P\left(X\cap Y\right)>0$, 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 $0$.