First slide

Theorems of probability

Question

$Events A and C are independent. If the probabilities relating A, B, and C are P (A) = 1 / 5, P (B) = 1 / 6; P (A \cap C) = \frac{1}{20}$
$P (B \cup C) = 3 / 8 . Then$

Moderate

A

events B and C arc independent
B

events B and C arc mutually exclusive
C

events B and C arc neither independent nor mutually exclusive
D

events B and C are equiprobable

Solution

$\begin{aligned} P (A \cap C) = P (A) P (C) \\ or \frac{1}{20} = \frac{1}{5} P (C) \\ or P (C) = \frac{1}{4} \end{aligned}$
now,
$\begin{aligned} P (B \cup C) = \frac{1}{6} + \frac{1}{4} - P (B \cap C) \\ P (B \cap C) = \frac{5}{12} - \frac{3}{8} = \frac{1}{24} = P (B) P (C) \end{aligned}$
$Therefore, B and C are independent.$

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