The probability of Jide, Atu and Obu solving a given problem are \(\frac{1}{12}\), \(\frac{1}{6}\) and \(\frac{1}{8}\) respectively. Calculate the probabili...

The probability of Jide, Atu and Obu solving a given problem are \(\frac{1}{12}\), \(\frac{1}{6}\) and \(\frac{1}{8}\) respectively. Calculate the probability that only one solves the problem.

Answer Details

To calculate the probability that only one person solves the problem, we can consider the following cases: Case 1: Jide solves the problem, and Atu and Obu don't solve the problem. The probability of Jide solving the problem is \(\frac{1}{12}\), the probability of Atu not solving the problem is \(\frac{5}{6}\) (since the probability of Atu solving the problem is \(\frac{1}{6}\)), and the probability of Obu not solving the problem is \(\frac{7}{8}\) (since the probability of Obu solving the problem is \(\frac{1}{8}\)). Therefore, the probability of this case is: \[\frac{1}{12} \times \frac{5}{6} \times \frac{7}{8} = \frac{35}{576}\] Case 2: Atu solves the problem, and Jide and Obu don't solve the problem. The probability of Atu solving the problem is \(\frac{1}{6}\), the probability of Jide not solving the problem is \(\frac{11}{12}\) (since the probability of Jide solving the problem is \(\frac{1}{12}\)), and the probability of Obu not solving the problem is \(\frac{7}{8}\). Therefore, the probability of this case is: \[\frac{1}{6} \times \frac{11}{12} \times \frac{7}{8} = \frac{77}{576}\] Case 3: Obu solves the problem, and Jide and Atu don't solve the problem. The probability of Obu solving the problem is \(\frac{1}{8}\), the probability of Jide not solving the problem is \(\frac{11}{12}\), and the probability of Atu not solving the problem is \(\frac{5}{6}\). Therefore, the probability of this case is: \[\frac{1}{8} \times \frac{11}{12} \times \frac{5}{6} = \frac{55}{576}\] So the total probability of only one person solving the problem is the sum of the probabilities of these three cases: \[\frac{35}{576} + \frac{77}{576} + \frac{55}{576} = \frac{167}{576}\] Therefore, the correct option is (d) \(\frac{167}{576}\).