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If $P(X)=0.33, P(Y)=0.75$, and $P(X \cap Y)=0.30$, are $X$ and $Y$ independent?
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Events $X$ and $Y$ are not independent.

Explanation:

If $X$ and $Y$ are independent event then:

$P(X) \cdot P(Y) = P(X\cap Y)$

Given that $P(X)=0.33, P(Y)=0.75$, and $P(X \cap Y)=0.30$, then:

$P(X) \cdot P(Y) =0.33\cdot 0.75 = 0.2475$, but  $P(X \cap Y)=0.30$

$\therefore P(X) \cdot P(Y) \neq P(X\cap Y)$

hence events $X$ and $Y$ are not independent.

by Diamond (40,719 points)

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