Probability
Conditional Probability and Independence
Grade 12
Question:
<p>If \(A\) and \(B\) are mutually exclusive events with \(P(A) = 0.5\), \(P(B) = 0.3\), which are TRUE?</p>
<p>\(P(A \cup B) = 0.8\)</p>
<p>\(P(A \cap B) = 0\)</p>
<p>\(A\) and \(B\) are independent</p>
<p>\(P(\bar{A} \cap \bar{B}) = 0.2\)</p>
Step-by-Step Solution
Key Concept: Mutually exclusive ⟹ P(A\capB)=0. Independent requires P(A\capB)=P(A)P(B)=0.15\neq0, so not independent.
<p><strong>A:</strong> \(P(A\cup B)=0.5+0.3=0.8\) ✓.</p><p><strong>B:</strong> Mutually exclusive → \(P(A\cap B)=0\) ✓.</p><p><strong>C:</strong> \(P(A)P(B)=0.15\neq0=P(A\cap B)\) → not independent. ✗.</p><p><strong>D:</strong> \(P(\bar A\cap\bar B)=1-0.8=0.2\) ✓.</p><p>Answer: ABD ✓.</p>
Correct Answer: ABD