QJEEAdv11Ma24c9: JEE Advanced 2011 Mathematics - Class 12 JEEAdv11Ma24c9 Solution: Let and be two independent events. The probability that exactly …

Let $P(E) = e$ and $P(F) = f$ .

$P(E \cup F) - P(E \cap F) = {{11} \over {25}}$

$\Rightarrow e + f - 2ef = {{11} \over {25}}$ ...... (1)

$P(\overline E \cap \overline F ) = {2 \over {25}}$

$\Rightarrow (1 - e)(1 - f) = {2 \over {25}}$

$\Rightarrow 1 - e - f + ef = {2 \over {25}}$ ...... (2)

From Eqs. (1) and (2), we get

$ef = {{12} \over {25}}$ and $e + f = {7 \over 5}$

Solving, we get

$e = {4 \over 5},\,f = {3 \over 5}$ or $e = {3 \over 5},\,f = {4 \over 5}$