Question

Suppose $E_1,E_2,E_3$ be three mutually exclusive events such that $P\left(E_i\right)=p_i$, for $i=1,2,3$.

If $p_1=\frac{1}{2}(1-p),p_2=\frac{1}{3}(1+2p)$ and $p_3=\frac{1}{5}(2+3p)$, then $p$ belongs to,

• A. $(-1,2)$
• B. $(-\frac{1}{3},\frac{2}{3})$
• C. $(-1,\frac{4}{7})$
• D. $[-\frac{1}{2},-\frac{7}{23}]$

Answer 1

Answer: (D)

$[-\frac{1}{2},-\frac{7}{23}]$

For mutually exclusive events,

$p_1\ge 0,p_2\ge 0,p_3\ge 0$ and $p_1+p_2+p_3\le 1$

$\Rightarrow p\le 1,p\ge \frac{-1}{2},p\ge \frac{-2}{3}$ and $p\le \frac{-7}{23}$

$\therefore \frac{-1}{2}\le p\le \frac{-7}{23}$