Question

If \( A \) and \( B \) are independent events such that odds in favour of \( A \) is 2:3 and odds against \( B \) is 4:5, then \( P(A \cap B) = \)

• A. \( \frac{1}{9} \)
• B. \( \frac{4}{5} \)
• C. \( \frac{2}{9} \)
• D. \( \frac{3}{9} \)

Hint:

For independent events, \( P(A \cap B) = P(A) \times P(B) \).

Answer 1

The Correct Option is C

Step 1: Convert odds to probabilities.
The odds in favour of \( A \) are 2:3, so \( P(A) = \frac{2}{5} \). The odds against \( B \) are 4:5, so \( P(B) = \frac{1}{5} \).
Step 2: Use the independence of events.
Since \( A \) and \( B \) are independent events, \( P(A \cap B) = P(A) \times P(B) \). Thus: \[ P(A \cap B) = \frac{2}{5} \times \frac{1}{5} = \frac{2}{25} = \frac{2}{9} \]
Step 3: Conclusion.
Thus, \( P(A \cap B) = \frac{2}{9} \), corresponding to option (C).