First slide

Theorems of probability

Question

$Let A and B be two events such that P (\bar{A \cup B}) = 1 / 6, P (A \cap B) = 1 / 4 and P (\bar{A}) = 1 / 4, where \bar{A} stands for$
$complement of event A . Then events A and B are$

Moderate

A

equally likely but not independent
B

equally likely and mutually exclusive
C

mutually exclusive and independent
D

independent but not equally likely

Solution

We have,
$\begin{aligned} P (\bar{A \cup B}) = \frac{1}{6}, P (A \cap B) = \frac{1}{4} \\ P (\bar{A}) = \frac{1}{4} \Rightarrow P (A) = 1 - \frac{1}{4} = \frac{3}{4} \end{aligned}$
$\begin{aligned} ∴ P (\vec{A \cup B}) = 1 - P (A \cup B) \\ = 1 - [P (A) + P (B) - P (A \cap B)] \end{aligned}$
$\begin{array}{l} \Rightarrow \frac{1}{6} = 1 - \frac{3}{4} - P (B) + \frac{1}{4} \\ \Rightarrow P (B) = 1 - \frac{1}{2} - \frac{1}{6} = \frac{6 - 3 - 1}{6} = \frac{2}{6} = \frac{1}{3} \\ \Rightarrow P (A \cap B) = P (A) P (B) = \frac{1}{4} \end{array}$

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