If $A,B,$ and $C$ are three events in a probability space such that $A$ and $B$ are independent, as well as $A$ and $C$ are independent, is $A$ independent of $B \cap C$? Explain.
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To answer this question, show your understanding of the definition of independence between events in probability. Remember that for two events, X and Y, to be independent, the probability of their intersection (P(X \\cap Y)) must equal the product of their probabilities (P(X) * P(Y)). Also, consider the implications of given independent pairs when assessing the independence of A from the intersection B \\cap C.
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