Probability
Classical Probability
Grade None
Question:
<p>A card is drawn from a well-shuffled pack of 52. \(A\) = card is a spade, \(B\) = card is a king. Which are TRUE?</p>
<p>\(A\) and \(B\) are independent</p>
<p>\(P(A|B) = \dfrac{1}{4}\)</p>
<p>\(P(A \cap B) = \dfrac{1}{52}\)</p>
<p>\(P(A \cup B) = \dfrac{4}{13}\)</p>
Step-by-Step Solution
Key Concept: P(A)=13/52=1/4, P(B)=4/52=1/13, P(A\capB)=1/52. Check: P(A)P(B)=1/52=P(A\capB) \to independent.
<p>\(P(A)=\frac{1}{4}\), \(P(B)=\frac{1}{13}\), \(P(A\cap B)=\frac{1}{52}\).</p><p><strong>A:</strong> \(P(A)P(B)=\frac{1}{4}\cdot\frac{1}{13}=\frac{1}{52}=P(A\cap B)\) → independent ✓.</p><p><strong>B:</strong> \(P(A|B)=\frac{1/52}{1/13}=\frac{1}{4}\) ✓.</p><p><strong>C:</strong> \(P(A\cap B)=\frac{1}{52}\) ✓.</p><p><strong>D:</strong> \(P(A\cup B)=\frac{1}{4}+\frac{1}{13}-\frac{1}{52}=\frac{13+4-1}{52}=\frac{16}{52}=\frac{4}{13}\) ✓.</p>
Correct Answer: ABCD