Question

If $\dfrac{{1 + 3p}}{3},\dfrac{{1 - p}}{4}$ and $\dfrac{{1 - 2p}}{2}$ are mutually exclusive events. Then, range of $p$ is
$\left( A \right)\dfrac{1}{3} \leqslant p \leqslant \dfrac{1}{2}$
$\left( B \right)\dfrac{1}{2} \leqslant p \leqslant \dfrac{1}{2}$
$\left( C \right)\dfrac{1}{3} \leqslant p \leqslant \dfrac{2}{3}$
$\left( D \right)\dfrac{1}{3} \leqslant p \leqslant \dfrac{2}{5}$

Answer 1

Hint : When two events $A$ and $B$ are Mutually Exclusive it is impossible for them to happen together: The probability of $A$ and $B$ together equals $0$ (impossible).
But, for Mutually Exclusive events, the probability of $A$ and $B$ is the sum of the individual probabilities: The probability of $A$ and $B$ equals the probability of $A$ plus the probability of $B$ .

Complete step-by-step answer :
To find the range of $p$ ,
Since, the probability lies between $0$ and $1$ ,
$0 \leqslant$ $\dfrac{{1 + 3p}}{3} \leqslant 1,0 \leqslant \dfrac{{1 - p}}{4} \leqslant 1,0 \leqslant$ $\dfrac{{1 - 2p}}{2} \leqslant 1$
That gives,
$\Rightarrow 0 \leqslant 1 + 3p \leqslant 3,0 \leqslant 1 - p \leqslant 4,0 \leqslant 1 - 2p \leqslant 2$
$\Rightarrow - \dfrac{1}{3} \leqslant p \leqslant \dfrac{2}{3}, - 3 \leqslant p \leqslant 1, - \dfrac{1}{2} \leqslant p \leqslant \dfrac{1}{2}$ _ _ _ _ _ _ _ _ _ _ $\left( 1 \right)$
But, when the events are mutually exclusive,
$\Rightarrow 0 \leqslant \dfrac{{1 + 3p}}{3} + \dfrac{{1 - p}}{4} + \dfrac{{1 - 2p}}{2} \leqslant 1$
$\Rightarrow 0 \leqslant 13 - 3p \leqslant \dfrac{{13}}{3}$ _ _ _ _ _ _ _ _ _ _ $\left( 2 \right)$
From equations $\left( 1 \right)$ and $\left( 2 \right)$ ,
\max \left\\{ { - \dfrac{1}{3}, - 3, - \dfrac{1}{2},\dfrac{1}{3}} \right\\} \leqslant p \leqslant \min \left\\{ {\dfrac{2}{3},1,\dfrac{1}{2},\dfrac{{13}}{3}} \right\\}
$\Rightarrow \dfrac{1}{3} \leqslant p \leqslant \dfrac{1}{2}.$
Therefore, the range of $p$ is $\left( A \right)\dfrac{1}{3} \leqslant p \leqslant \dfrac{1}{2}$ .
So, the correct answer is “Option A”.

Note : $\Rightarrow$ In logic and probability theory, two events are mutually exclusive or disjoint if they cannot both occur at the same time.
$\Rightarrow$ A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both.