Question

The probability of \( A \) winning the race is \( \frac{1}{3} \) and that of \( B \) is \( \frac{1}{4} \). In this race, find the probability that neither \( A \) nor \( B \) can win the race.

Hint:

For probability of "neither event happening," use \( P(\text{Neither}) = 1 - P(A \cup B) \). If events are independent, use \( P(A \cap B) = P(A) \times P(B) \).

Answer 1

Step 1: Given probabilities. Let \( P(A) = \frac{1}{3} \) and \( P(B) = \frac{1}{4} \). 

Step 2: Find the probability that at least one wins. Using the formula for union of probabilities: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since nothing is mentioned about \( A \) and \( B \) winning together, we assume independence: \[ P(A \cap B) = P(A) \times P(B) = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12} \] 

Step 3: Compute \( P(A \cup B) \). \[ P(A \cup B) = \frac{1}{3} + \frac{1}{4} - \frac{1}{12} \] Finding LCM of 3, 4, and 12: \[ P(A \cup B) = \frac{4}{12} + \frac{3}{12} - \frac{1}{12} = \frac{6}{12} = \frac{1}{2}). \] 

Step 4: Find probability that neither wins. \[ P(\text{Neither } A \text{ nor } B) = 1 - P(A \cup B) = 1 - \frac{1}{2} = \frac{1}{2} \] Thus, the probability that neither wins is \( \frac{1}{2} \