Probability
Mutually Exclusive Events
Grade 12
Question:
<p>For mutually exclusive events: \(P(A \cup B) = P(A) + P(B)\) and \(P(A \cap B) = 0\). Consider the experiment of throwing a die. Let \(A\) = event that the number obtained is odd and \(B\) = event that the number obtained is even. Which of the following statements is correct?</p>
<p>\(P(A) = P(\bar{B})\), \(P(B) = P(\bar{A})\) and \(P(A) = P(B)\)</p>
<p>Options (1), (2) and (3) are all correct</p>
<p>\(P(A) + P(B) = 1\)</p>
<p>Options (1), (2) and (3) are all incorrect</p>
Step-by-Step Solution
Key Concept: Mutually exclusive events have zero intersection and their union probability equals the sum of individual probabilities. Events A (odd) and B (even) on a die are mutually exclusive because no outcome can be both odd and even simultaneously.
<p><strong>Step 1:</strong> Identify the sample space for a die: S = {1, 2, 3, 4, 5, 6}</p><p><strong>Step 2:</strong> Define the events:</p><ul><li>A (odd numbers) = {1, 3, 5}, so P(A) = 3/6 = 1/2</li><li>B (even numbers) = {2, 4, 6}, so P(B) = 3/6 = 1/2</li></ul><p><strong>Step 3:</strong> Verify mutual exclusivity:</p><ul><li>A ∩ B = ∅ (empty set) because no number is both odd and even</li><li>Therefore, P(A ∩ B) = 0 ✓</li></ul><p><strong>Step 4:</strong> Calculate union:</p><ul><li>P(A ∪ B) = P(A) + P(B) = 1/2 + 1/2 = 1</li><li>This makes sense: every outcome on the die is either odd or even</li></ul><p><strong>Step 5:</strong> The correct statement is: A and B are mutually exclusive events with P(A ∩ B) = 0 and P(A ∪ B) = 1.</p><p>∴ Answer: D</p>
Correct Answer: D