First slide

Introduction to probability

Question

If the events A and B arc mutually exclusive events such that $P (A) = \frac{3 x + 1}{3} and P (B) = \frac{1 - x}{4}$ , then the set of possible real values of x lies in the interval

Moderate

A

$[0, 1]$
B

$[- \frac{1}{3}, \frac{5}{9}]$
C

$[- \frac{7}{9}, \frac{4}{9}]$
D

$[\frac{1}{3}, \frac{2}{3}]$

Solution

We must have,
$\begin{array}{l} 0 \leq P (A) \leq 1 \\ \Rightarrow 0 \leq \frac{3 x + 1}{3} \leq 1 \\ \Rightarrow \frac{- 1}{3} \leq x \leq \frac{2}{3} (1) \end{array}$
and $0 \leq P (B) \leq 1$
$\Rightarrow 0 \leq \frac{1 - x}{4} \leq 1$
$\Rightarrow - 3 \leq x \leq 1$ (2)
and $0 \leq P (A \cup B) = P (A) + P (B) \leq 1$
$\Rightarrow 0 \leq \frac{3 x + 1}{3} + \frac{1 - x}{4} \leq 1$
$\Rightarrow \frac{- 7}{9} \leq x \leq \frac{5}{9}$ (3)
From (1), (2) and (3), we get $x \in [\frac{- 1}{3}, \frac{5}{9}]$ .

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