Probability
Conditional Probability
Grade 12
Question:
<p>If \(A\) and \(B\) are any two events such that \(P(A) = 2/5\) and \(P(A \cap B) = 3/20\), then the conditional probability, \(P(A \mid (A' \cup B'))\), where \(A'\) denotes the complement of \(A\), is equal to</p>
<p>\(11/20\)</p>
<p>\(5/17\)</p>
<p>\(8/17\)</p>
<p>\(1/4\)</p>
Step-by-Step Solution
Key Concept: Use the formula P(A|(A'∪B')) = P(A∩(A'∪B'))/P(A'∪B'). Notice that A∩(A'∪B') = ∅ (empty set) since A and A' are mutually exclusive, making the numerator zero regardless of B.
<p><strong>Step 1:</strong> Recall that for conditional probability: P(A|(A'∪B')) = P(A∩(A'∪B'))/P(A'∪B')</p><p><strong>Step 2:</strong> Find A∩(A'∪B'). By distributive property: A∩(A'∪B') = (A∩A')∪(A∩B'). Since A∩A' = ∅, we have A∩(A'∪B') = ∅</p><p><strong>Step 3:</strong> Therefore P(A∩(A'∪B')) = P(∅) = 0</p><p><strong>Step 4:</strong> Thus P(A|(A'∪B')) = 0/P(A'∪B') = 0 (provided P(A'∪B') ≠ 0, which is true since A'∪B' includes all elements outside A)</p><p>∴ Answer: <strong>0 (Option C)</strong></p>
Correct Answer: C