Question

A problem is given to 3 students whose chances of solving it are \( \frac{1}{3} \), \( \frac{1}{5} \), and \( \frac{1}{6} \). What is the probability that:
(i) Exactly one of them solves the problem?

(ii) The problem is solved?

Hint:

In probability problems involving multiple events, remember to use the complement rule and the sum of probabilities for mutually exclusive events.

Answer 1

Step 1: Probability that exactly one student solves the problem.
Let \( P_1 = \frac{1}{3} \), \( P_2 = \frac{1}{5} \), and \( P_3 = \frac{1}{6} \) be the probabilities of the three students solving the problem. The probability that a student does not solve the problem is the complement of their solving probability: \[ P_1' = 1 - P_1 = \frac{2}{3}, \quad P_2' = 1 - P_2 = \frac{4}{5}, \quad P_3' = 1 - P_3 = \frac{5}{6} \] The probability that exactly one student solves the problem is the sum of the probabilities of each student solving the problem while the others do not. This can be expressed as: \[ P(\text{exactly one solves}) = P_1 \cdot P_2' \cdot P_3' + P_1' \cdot P_2 \cdot P_3' + P_1' \cdot P_2' \cdot P_3 \] Substitute the values of \( P_1 \), \( P_2 \), \( P_3 \), and their complements: \[ P(\text{exactly one solves}) = \frac{1}{3} \cdot \frac{4}{5} \cdot \frac{5}{6} + \frac{2}{3} \cdot \frac{1}{5} \cdot \frac{5}{6} + \frac{2}{3} \cdot \frac{4}{5} \cdot \frac{1}{6} \] Simplifying: \[ P(\text{exactly one solves}) = \frac{20}{90} + \frac{10}{90} + \frac{8}{90} = \frac{38}{90} = \frac{19}{45} \] Step 2: Probability that the problem is solved.
The probability that the problem is solved is the complement of the probability that none of the students solves the problem. The probability that none of the students solves the problem is: \[ P(\text{none solves}) = P_1' \cdot P_2' \cdot P_3' = \frac{2}{3} \cdot \frac{4}{5} \cdot \frac{5}{6} \] Simplifying: \[ P(\text{none solves}) = \frac{40}{90} = \frac{4}{9} \] Therefore, the probability that the problem is solved is: \[ P(\text{solved}) = 1 - P(\text{none solves}) = 1 - \frac{4}{9} = \frac{5}{9} \] Step 3: Conclusion.
(i) The probability that exactly one of the students solves the problem is \( \frac{19}{45} \).
(ii) The probability that the problem is solved is \( \frac{5}{9} \).