<p>We have \(A = \{4, 5, 6\}\), \(B = \{1, 2, 3, 4\}\). We have \(A \cup B = \{1, 2, 3, 4, 5, 6\} = S\) where \(S\) is the sample space of the experiment of throwing a die, so \(P(S) = 1\). Hence \(P(A \cup B) =\)?</p>
Step-by-Step Solution
Key Concept: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). First identify A ∩ B = {4}, then calculate probabilities based on the sample space S = {1,2,3,4,5,6}.
<p><strong>Step 1:</strong> Identify the sets and sample space.</p><p>A = {4, 5, 6}, B = {1, 2, 3, 4}, S = {1, 2, 3, 4, 5, 6} (die outcomes)</p><p><strong>Step 2:</strong> Find A ∩ B (common elements).</p><p>A ∩ B = {4}</p><p><strong>Step 3:</strong> Calculate individual probabilities.</p><p>P(A) = 3/6 = 1/2 (three favorable outcomes out of 6)</p><p>P(B) = 4/6 = 2/3 (four favorable outcomes out of 6)</p><p>P(A ∩ B) = 1/6 (one common element)</p><p><strong>Step 4:</strong> Apply union formula: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)</p><p>P(A ∪ B) = 1/2 + 2/3 − 1/6 = 3/6 + 4/6 − 1/6 = 6/6 = 1</p><p><strong>Verification:</strong> A ∪ B = {1,2,3,4,5,6} = S, so P(A ∪ B) = P(S) = 1 ✓</p><p>∴ Answer: <strong>1</strong></p>
Correct Answer: C