Question

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

• A. $[0,1]$
• B. $\left[\frac{-1}{3},\frac{5}{9}\right]$
• C. $\left[\frac{-7}{9},\frac{4}{9}\right]$
• D. $\left[\frac{1}{3},\frac{2}{3}\right]$

Answer 1

Answer: (B)

We must have,
$0\le P(A)\le 1$
$\Rightarrow \frac{-1}{3}\le x\le \frac{2}{3}\dots \dots$ (1)
$0\le P(B)\le 1$
$\Rightarrow -3\le x\le 1\dots \dots$ (2)
and $0\le P(A\cup B)$
$=P(A)+P(B)\le 1$
$\Rightarrow \frac{-7}{9}\le x\le \frac{5}{9}\dots \dots$ (3)
$\therefore (1)\cap (2)\cap (3)$
$\Rightarrow x\in \left[\frac{-1}{3},\frac{5}{9}\right]$