<p>In throwing of a die, let A be the event 'an odd number turns up', B be the event 'a number divisible by 3 turns up' and C be the event 'a number \(\le 4\) turns up'. The probability that exactly two of A, B and C occur is \(\frac{a}{b}\), then \(a + b\) is ……….</p>
Step-by-Step Solution
Key Concept: Find the probability that exactly two of three events occur by identifying favorable outcomes where exactly two events are satisfied.
<p><strong>Solution:</strong></p><p>Sample space: {1, 2, 3, 4, 5, 6}</p><p>Event A (odd): {1, 3, 5} → P(A) = 1/2</p><p>Event B (divisible by 3): {3, 6} → P(B) = 1/3</p><p>Event C (≤ 4): {1, 2, 3, 4} → P(C) = 2/3</p><p>Exactly two of A, B, C occur means:</p><p>Case 1: A and B occur, C does not</p><p>A ∩ B = {3}, A ∩ B ∩ C' = {3} (since 3 ≤ 4, this is empty) = ∅</p><p>Case 2: A and C occur, B does not</p><p>A ∩ C ∩ B' = {1} → Probability = 1/6</p><p>Case 3: B and C occur, A does not</p><p>B ∩ C ∩ A' = {6} ∩ {1,2,3,4} = ∅</p><p>Total probability = 1/6, so a = 1, b = 6</p><p>a + b = 7</p>
Correct Answer: 7