Probability that a Problem is Solved by at Least One Person

To find the probability that the problem is correctly solved by at least one of the four individuals, we can use the complement rule.

Let $A, B, C,$ and $D$ be the events that the first, second, third, and fourth individuals solve the problem correctly, respectively.

The given probabilities are: $P(A) = \frac{1}{2}$ $P(B) = \frac{3}{4}$ $P(C) = \frac{1}{4}$ $P(D) = \frac{1}{8}$

The probabilities that each individual fails to solve the problem are: $P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2}$ $P(B') = 1 - P(B) = 1 - \frac{3}{4} = \frac{1}{4}$ $P(C') = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4}$ $P(D') = 1 - P(D) = 1 - \frac{1}{8} = \frac{7}{8}$

Since the individuals solve the problem independently, the events $A', B', C',$ and $D'$ are also mutually independent. Thus, the probability that none of them solves the problem correctly is: $P(\text{none solves}) = P(A' \cap B' \cap C' \cap D') = P(A') \cdot P(B') \cdot P(C') \cdot P(D')$ $P(\text{none solves}) = \frac{1}{2} \times \frac{1}{4} \times \frac{3}{4} \times \frac{7}{8} = \frac{21}{256}$

Therefore, the probability that the problem is solved correctly by at least one of them is: $P(\text{at least one solves}) = 1 - P(\text{none solves}) = 1 - \frac{21}{256} = \frac{235}{256}$

Hence, the correct option is (A).