Let A, B, and C be three mutually independent | vedclass

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(A) Given that $A, B,$ and $C$ are mutually independent events,we have $P(A \cap B) = P(A)P(B)$,$P(A \cap C) = P(A)P(C)$,and $P(A \cap B \cap C) = P(A

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Both $S_1$ and $S_2$ are true.

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(A) Given that $A, B,$ and $C$ are mutually independent events,we have $P(A \cap B) = P(A)P(B)$,$P(A \cap C) = P(A)P(C)$,and $P(A \cap B \cap C) = P(A)P(B)P(C)$.For $S_1$: We check if $P(A \cap (B \cup C)) = P(A)P(B \cup C)$.$P(A \cap (B \cup C)) = P((A \cap B) \cup (A \cap C)) = P(A \cap B) + P(A \cap C) - P(A \cap B \cap C)$$= P(A)P(B) + P(A)P(C) - P(A)P(B)P(C) = P(A)[P(B) + P(C) - P(B)P(C)]$$= P(A)P(B \cup C)$. Thus,$S_1$ is true.For $S_2$: We check if $P(A \cap (B \cap C)) = P(A)P(B \cap C)$.$P(A \cap (B \cap C)) = P(A \cap B \cap C) = P(A)P(B)P(C) = P(A)P(B \cap C)$. Thus,$S_2$ is true.Therefore,both $S_1$ and $S_2$ are true.

Let $A, B,$ and $C$ be three mutually independent events. Consider the two statements $S_1$ and $S_2$:
$S_1: A$ and $B \cup C$ are independent.
$S_2: A$ and $B \cap C$ are independent.
Then:

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Let $A, B,$ and $C$ be three mutually independent events. Consider the two statements $S_1$ and $S_2$:$S_1: A$ and $B \cup C$ are independent.$S_2: A$ and $B \cap C$ are independent.Then: