Let A, B, C be three mutually independent events. Consider the two statements S1 and S2 S1: A and B∪C are independentS2: A and B∩C are independentThen,

We have PA∩B∪C=PA∩B∪A∩C=PA∩B+PA∩C−PA∩B∩C=pAPB+PAPC−PAPBPC=PAPB+PC−PB∩C=PAPB∪C∴ S1 is true.PA∩B∩C=PAPBPC=PAPB∩C∴ S2 is also true.

Let A, B, C be three mutually independent events. Consider the two statements S1 and S2 S1: A and B∪C are independentS2: A and B∩C are independentThen,

We have PA∩B∪C=PA∩B∪A∩C=PA∩B+PA∩C−PA∩B∩C=pAPB+PAPC−PAPBPC=PAPB+PC−PB∩C=PAPB∪C∴ S1 is true.PA∩B∩C=PAPBPC=PAPB∩C∴ S2 is also true.

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Let A, B, 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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Both $S_{1}$ and $S_{2}$ are true

Only $S_{1}$ is true

Only $S_{2}$ true

Neither $S_{1}$ nor $S_{2}$ is true

answer is A.

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Detailed Solution

We have $P [A \cap (B \cup C)] = P [(A \cap B) \cup (A \cap C)]$

$= P (A \cap B) + P (A \cap C) - P (A \cap B \cap C)$

$= p (A) P (B) + P (A) P (C) - P (A) P (B) P (C)$

$= P (A) [P (B) + P (C) - P (B \cap C)]$

$= P (A) P (B \cup C)$

$∴ S_{1}$ is true.

$P (A \cap (B \cap C)) = P (A) P (B) P (C) = P (A) P (B \cap C)$

$∴ S_{2}$ is also true.

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Let A, B, C be three mutually independent events. Consider the two statements S1 and S2 S1: A and B∪C are independentS2: A and B∩C are independentThen,

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Let A, B, 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,