AP DSC School Assistant Math Model Test 3

Concept:The problem uses the concept of independent events. The probability that at least one student solves the problem is found by first calculating the probability that none of them solves it (i.e., all fail), and then subtracting that from 1.Explanation:Let A, B, C denote the events that students A, B, C solve the problem. Their probabilities are given:$P(A) = \frac{1}{2},\quad P(B) = \frac{1}{3},\quad P(C) = \frac{1}{4}$First, find the probabilities that each student fails (complement events):$P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2}$$P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3}$$P(C') = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4}$Since the students try independently, the probability that none of them solves the problem (i.e., all three fail) is the product of their individual failure probabilities:$P(\text{not solved}) = P(A' \cap B' \cap C') = P(A') \cdot P(B') \cdot P(C') = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} = \frac{1}{4}$Therefore, the probability that the problem is solved (at least one succeeds) is:$P(\text{solved}) = 1 - P(\text{not solved}) = 1 - \frac{1}{4} = \frac{3}{4}$Answer:The probability that the problem is solved is $\frac{3}{4}$, which corresponds to option B.