JEE MAIN - Mathematics (2022 - 26th July Morning Shift - No. 13)
Let $$\mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{E}_{3}$$ be three mutually exclusive events such that $$\mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{2+3 \mathrm{p}}{6}, \mathrm{P}\left(\mathrm{E}_{2}\right)=\frac{2-\mathrm{p}}{8}$$ and $$\mathrm{P}\left(\mathrm{E}_{3}\right)=\frac{1-\mathrm{p}}{2}$$. If the maximum and minimum values of $$\mathrm{p}$$ are $$\mathrm{p}_{1}$$ and $$\mathrm{p}_{2}$$, then $$\left(\mathrm{p}_{1}+\mathrm{p}_{2}\right)$$ is equal to :
$$\frac{2}{3}$$
$$\frac{5}{3}$$
$$\frac{5}{4}$$
1
Explanation
$$0 \le {{2 + 3P} \over 6} \le 1 \Rightarrow P \in \left[ { - {2 \over 3},{4 \over 3}} \right]$$
$$0 \le {{2 - P} \over 8} \le 1 \Rightarrow P \in [ - 6,2]$$
$$0 \le {{1 - P} \over 2} \le 1 \Rightarrow P \in [ - 1,1]$$
$$0 < P({E_1}) + P({E_2}) + P({E_3}) \le 1$$
$$0 < {{13} \over {12}} - {P \over 8} \le 1$$
$$P \in \left[ {{2 \over 3},{{26} \over 3}} \right]$$
Taking intersection of all
$$P \in \left[ {{2 \over 3},1} \right)$$
$${P_1} + {P_2} = {5 \over 3}$$
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