Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability of none of them occurring is $\frac{2}{25}$. If $P(T)$ denotes the probability of occurrence of the event $T$, then

$E,\ F \text{ independent}$

$P(\text{exactly one}) = \frac{11}{25}$

$P(\text{none}) = \frac{2}{25}$

$\text{To find: } P(E),\ P(F)$

$\text{Let } p = P(E), \quad q = P(F)$

$(1-p)(1-q) = \frac{2}{25}$

$p(1-q) + q(1-p) = \frac{11}{25}$

$P(E \cap F) = 1 - \frac{11}{25} - \frac{2}{25} = \frac{12}{25}$

$P(E \cap F) = pq = \frac{12}{25}$

$p + q - 2pq = \frac{11}{25}$

$p + q = \frac{11}{25} + 2 \cdot \frac{12}{25} = \frac{35}{25} = \frac{7}{5}$

$t^2 - \frac{7}{5}\,t + \frac{12}{25} = 0$

$t = \frac{\frac{7}{5} \pm \frac{1}{5}}{2} = \frac{4}{5},\ \frac{3}{5}$

$P(E)=\frac{4}{5}, \quad P(F)=\frac{3}{5} \ \Rightarrow \ \text{option (1)}$