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.