Four persons independently solve a certain problem correctly with probabilities $\frac{1}{2}, \frac{3}{4}, \frac{1}{4}, \frac{1}{8}$. Then the probability that the problem is solved correctly by at least one of them is

$P_1 = \frac{1}{2},\quad P_2 = \frac{3}{4},\quad P_3 = \frac{1}{4},\quad P_4 = \frac{1}{8}$

$\text{Each solves independently}$

$\text{To find: } P(\text{at least one solves})$

$= 1 - P(\text{none solves})$

$\text{Plan: } 1.\ \bar{P}_i = 1 - P_i$

$2.\ P(\text{none}) = \bar{P}_1\bar{P}_2\bar{P}_3\bar{P}_4 \ (\text{independent})$

$3.\ P(\ge 1) = 1 - P(\text{none})$

$\bar{P}_i = 1 - P_i$

$\bar{P}_1=\frac{1}{2},\quad \bar{P}_2=\frac{1}{4},\quad \bar{P}_3=\frac{3}{4},\quad \bar{P}_4=\frac{7}{8}$

$P(\text{none solves}) = \bar{P}_1 \times \bar{P}_2 \times \bar{P}_3 \times \bar{P}_4$

$= \frac{1}{2} \times \frac{1}{4} \times \frac{3}{4} \times \frac{7}{8}$

$= \frac{1 \times 1 \times 3 \times 7}{2 \times 4 \times 4 \times 8}$

$= \frac{21}{256}$

$P(\text{at least one}) = 1 - \frac{21}{256}$

$= \frac{256 - 21}{256} = \frac{235}{256}$

$P(\text{at least one solves}) = \frac{235}{256}$

$\text{Correct option: } (1)$