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14.Probability
hard
Let $E _{1}, E _{2}, E _{3}$ be three mutually exclusive events such that $P \left( E _{1}\right)=\frac{2+3 p }{6}, P \left( E _{2}\right)=\frac{2- p }{8}$ and $P \left( E _{3}\right)$ $=\frac{1- p }{2}$. If the maximum and minimum values of $p$ are $p _{1}$ and $p _{2}$, then $\left( p _{1}+ p _{2}\right)$ is equal to.
A
$\frac{2}{3}$
B
$\frac{5}{3}$
C
$\frac{5}{4}$
D
$1$
(JEE MAIN-2022)
Solution
$0 \leq P \left( E _{ i }\right) \leq 1$ for $i =1,2,3$
$-2 / 3 \leq p \leq 1$
$E _{1},E _{2},E _{3}$ are mutually exclusive
$P \left( E _{1}\right)+ P \left( E _{2}\right)+ P _{\left( E _{3}\right)} \leq 1$
$2 / 3 \leq p \leq 1$
$p _{1}=1, p _{2}=2 / 3$
$p _{1}+ p _{2}=5 / 3$
Standard 11
Mathematics
