First slide

Theorems of probability

Question

Let A, B, C be three mutually independent events. Consider the two statements S₁ and S₂.
S₁: A and $B \cup C$ are independent
S₂: A and $B \cap C$ are independent
Then,

Moderate

A

$both S_{1} and S_{2} are true$
B

$only S_{1} is true$
C

$only S_{2} is true$
D

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

Solution

$\begin{matrix} P [A \cap (B \cup C)] = P [(A \cap B) \cup (A \cap C)] \\ = P (A \cap B) + P (A \cap C) - P (A \cap B \cap C) \\ = P (A) P (B) + P (A) P (C) - P (A) P (B) P (C) \\ = P (A) [P (B) + P (C) - P (B \cap C)] \\ = P (A) P (B \cup C) \end{matrix}$
$Therefore, S_{1} is true.$
$P (A \cap (B \cap C)) = P (A) P (B) P (C) = P (A) P (B \cap C)$
$Therefore, S_{2} is also true.$

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