Probability
Event and Algebra of Events
Grade 12
Question:
<p>If <i>A</i> and <i>B</i> are two events, the probability that exactly one of them occurs is given by</p>
<p>(1) \(P(A) + P(B) - 2P(A \cap B)\)</p>
<p>(2) \(P(A \cap \bar{B}) + P(\bar{A} \cap B)\)</p>
<p>(3) \(P(A \cup B) - P(A \cap B)\)</p>
<p>(4) \(P(\bar{A}) + P(B) - 2P(\bar{A} \cap B)\)</p>
Step-by-Step Solution
Key Concept: The probability that exactly one event occurs equals P(A∪B) - P(A∩B), which can be rewritten as P(A) + P(B) - 2P(A∩B) or equivalently P(A) + P(B) - P(A∩B) - P(A∩B). This represents the symmetric difference of the two events.
<p><strong>Step 1:</strong> Identify what 'exactly one occurs' means: either A occurs and B doesn't, or B occurs and A doesn't.</p><p><strong>Step 2:</strong> Express this as: P(exactly one) = P(A∩B') + P(A'∩B)</p><p><strong>Step 3:</strong> Expand using complement rule: = P(A)(1-P(B)) + P(B)(1-P(A))</p><p><strong>Step 4:</strong> Simplify: = P(A) + P(B) - 2P(A∩B)</p><p><strong>Alternative form:</strong> P(A∪B) - P(A∩B) = P(A) + P(B) - P(A∩B) - P(A∩B)</p><p>∴ Answer: <strong>P(A) + P(B) - 2P(A∩B)</strong> or equivalently <strong>P(A∪B) - P(A∩B)</strong></p>
Correct Answer: 1,2,3,4