First slide

Introduction to probability

Question

Given two events A and B, if the odds against A are 2 to 1 those in favour of AUB are 3 to 1, then

Moderate

A

$\frac{1}{3} \leq P (B) \leq \frac{1}{2}$
B

$\frac{1}{2} \leq P (B) \leq \frac{3}{4}$
C

$\frac{5}{12} \leq P (B) \leq \frac{3}{4}$
D

$0 \leq P (B) \leq 1$

Solution

Given that
$\begin{array}{l} P (\bar{A}) : P (A) = 2 : 1 \Rightarrow P (\bar{A}) = \frac{2}{3}, P (A) = \frac{1}{3} \\ P (A \cup B) : P (\bar{A \cup B}) = 3 : 1 \\ \Rightarrow P (A \cup B) = \frac{3}{4}, P (\bar{A \cup B}) = \frac{1}{4} \\ P (A \cup B) - P (A) \leq P (B) \leq P (A \cup B) \\ \Rightarrow \frac{5}{12} \leq P (B) \leq \frac{3}{4} \end{array}$

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