Question

Given two events \( A \) and \( B \) such that the odds in favour of \( A \) are 2:1 and the odds in favour of \( A \cup B \) are 3:1. Consistent with this information, the smallest and largest value for the probability of event \( B \) are given by

• A. \( \frac{1}{12} \leq P(B) \leq \frac{3}{4} \)
• B. \( \frac{1}{3} \leq P(B) \leq \frac{1}{2} \)
• C. \( \frac{1}{6} \leq P(B) \leq \frac{1}{3} \)
• D. None of these

Hint:

In probability problems involving odds, first convert the odds to probabilities, and then use the formula for the union of events to calculate the desired probabilities.

Answer 1

The Correct Option is A

We are given the following information: - The odds in favour of \( A \) are 2:1, so the probability of event \( A \), \( P(A) \), is: \[ P(A) = \frac{2}{3} \] - The odds in favour of \( A \cup B \) are 3:1, so the probability of event \( A \cup B \), \( P(A \cup B) \), is: \[ P(A \cup B) = \frac{3}{4} \] We can use the formula for the probability of the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substitute the known values: \[ \frac{3}{4} = \frac{2}{3} + P(B) - P(A \cap B) \] Now, we use the fact that the probability of the intersection \( P(A \cap B) \) is bounded by \( P(A) \) and \( P(B) \). Since the probability of an intersection cannot exceed the smaller of the probabilities of \( A \) and \( B \), the value of \( P(A \cap B) \) can range from 0 to \( \min(P(A), P(B)) \). Thus, we analyze the range for \( P(B) \): 

1. When \( P(A \cap B) = 0 \), the equation becomes: \[ \frac{3}{4} = \frac{2}{3} + P(B) \] Solving for \( P(B) \), we get: \[ P(B) = \frac{3}{4} - \frac{2}{3} = \frac{9}{12} - \frac{8}{12} = \frac{1}{12} \] 2. When \( P(A \cap B) = P(B) \), the equation becomes: \[ \frac{3}{4} = \frac{2}{3} + P(B) - P(B) \] Simplifying, we get: \[ P(B) = \frac{3}{4} - \frac{2}{3} = \frac{9}{12} - \frac{8}{12} = \frac{3}{4} \] Thus, the smallest value of \( P(B) \) is \( \frac{1}{12} \) and the largest value is \( \frac{3}{4} \). Therefore, the correct answer is \( \boxed{(a)} \).