Probability
Basic Probability
Grade 12
Question:
<p>The probability that exactly one of the two events \(A, B\) occurs is</p>
<p>(a) \(P(A) + P(B) - 2 \cdot P(A \cap B)\)</p>
<p>(b) \(P(A) + P(B) - P(A \cap B)\)</p>
<p>(c) \(P(A) + P(B) - 2 \cdot P(A \cap B)\)</p>
<p>(d) \(P(A \cap B) + P(A \cap B)\)</p>
Step-by-Step Solution
Key Concept: The event 'exactly one of A, B occurs' means A happens but B doesn't, OR B happens but A doesn't. This is the symmetric difference: (A ∩ B') ∪ (A' ∩ B), which equals P(A) + P(B) - 2P(A ∩ B) when events may be dependent.
<p><strong>Step 1:</strong> Define the event 'exactly one of A, B occurs' = (A ∩ B') ∪ (A' ∩ B)</p><p><strong>Step 2:</strong> Since these two cases are mutually exclusive: P(exactly one) = P(A ∩ B') + P(A' ∩ B)</p><p><strong>Step 3:</strong> Rewrite using complements: P(A ∩ B') = P(A) - P(A ∩ B) and P(A' ∩ B) = P(B) - P(A ∩ B)</p><p><strong>Step 4:</strong> Add them: P(exactly one) = P(A) - P(A ∩ B) + P(B) - P(A ∩ B) = P(A) + P(B) - 2P(A ∩ B)</p><p><strong>Alternative form:</strong> If A and B are independent: P(exactly one) = P(A)[1 - P(B)] + P(B)[1 - P(A)] = P(A) + P(B) - 2P(A)P(B)</p><p>∴ Answer: A</p>
Correct Answer: A