Question

The probabilities of solving a question by \( A \) and \( B \) independently are \( \frac{1}{2} \) and \( \frac{1}{3} \) respectively. If both of them try to solve it independently, find the probability that:

 

 

Hint:

For independent events, the probability that neither occurs is the product of their failure probabilities. Use the complement rule: \[ P(\text{at least one}) = 1 - P(\text{none}). \]

Answer 1

Step 1: Define probabilities. Let \( P(A) = \frac{1}{2} \) be the probability that \( A \) solves the problem. Let \( P(B) = \frac{1}{3} \) be the probability that \( B \) solves the problem. 

Step 2: Compute probability that neither solves it. \[ P(\text{none}) = P(\text{A fails}) \times P(\text{B fails}) \] Since the events are independent, \[ P(A'B') = (1 - P(A)) \times (1 - P(B)) \] \[ = \left( 1 - \frac{1}{2} \right) \times \left( 1 - \frac{1}{3} \right) \] \[ = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3} \] Thus, the probability that none of them solve it is \( \frac{1}{3} \). 

Step 3: Compute probability that at least one solves it. Using the complement rule: \[ P(\text{at least one}) = 1 - P(\text{none}) \] \[ = 1 - \frac{1}{3} = \frac{2}{3} \] Thus, the probability that at least one of them solves it is \( \frac{2}{3} \).