Verified Solution • 2025–26 Session moderate

QJEEM22Mae9b7: JEE Main 2022 Mathematics - Class 12 JEEM22Mae9b7 Solution: Let be three mutually exclusive events such that and . If the…

Let E1,E2,E3\mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{E}_{3} be three mutually exclusive events such that P(E1)=2+3p6,P(E2)=2−p8\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 P(E3)=1−p2\mathrm{P}\left(\mathrm{E}_{3}\right)=\frac{1-\mathrm{p}}{2} . If the maximum and minimum values of p\mathrm{p} are p1\mathrm{p}_{1} and p2\mathrm{p}_{2} , then (p1+p2)\left(\mathrm{p}_{1}+\mathrm{p}_{2}\right) is equal to :

Multiple Choice Options

  • A 23\frac{2}{3}
  • B 53\frac{5}{3}
  • C 54\frac{5}{4}
  • D 1

Step-by-Step Verified Solution

0≤2+3P6≤1⇒P∈[−23,43]0 \le {{2 + 3P} \over 6} \le 1 \Rightarrow P \in \left[ { - {2 \over 3},{4 \over 3}} \right]

0≤2−P8≤1⇒P∈[−6,2]0 \le {{2 - P} \over 8} \le 1 \Rightarrow P \in [ - 6,2]

0≤1−P2≤1⇒P∈[−1,1]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∈[23,263]P \in \left[ {{2 \over 3},{{26} \over 3}} \right]

Taking intersection of all

P∈[23,1)P \in \left[ {{2 \over 3},1} \right)

P1+P2=53{P_1} + {P_2} = {5 \over 3}

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