Question

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

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

Answer 1

The correct option is C $[\frac{-1}{3},\frac{5}{9}]$


We know that
$0\le P\left(A\right)\le 1,0\le P\left(B\right)\le 1$ and $0\le P\left(A\right)+P\left(B\right)\le 1$
$\Rightarrow 0\le \frac{3x+1}{3}\le 1,0\le \frac{1-x}{4}\le 1,0\le \frac{3x+1}{3}+\frac{1-x}{4}\le 1\Rightarrow 0\le 3x+1\le 3,0\le 1-x\le 4,0\le 9x+7\le 12\Rightarrow -\frac{1}{3}\le x\le \frac{2}{3},-3\le x\le 1,-\frac{7}{9}\le x\le \frac{5}{9}\Rightarrow x\in [\frac{-1}{3},\frac{5}{9}]$