Probability
Classical Probability
Grade 12
Question:
<p>A bag contains 5 black and 3 red balls. Two balls drawn without replacement. \(A\)=first is black, \(B\)=second is red. Which are TRUE?</p>
P(A) = 5/8
P(B) = 3/8
P(A \cap B) = 15/56
A and B are independent
Step-by-Step Solution
Key Concept: P(A) = 5/8 by symmetry. P(B) = 3/8 by symmetry (each ball equally likely to be second). P(A\capB) = (5/8)(3/7) = 15/56.
<p>\(P(A)=\frac{5}{8}\). ✓ <strong>A</strong>.</p><p>\(P(B)=P(B|A)P(A)+P(B|\bar A)P(\bar A)=\frac{3}{7}\cdot\frac{5}{8}+\frac{2}{7}\cdot\frac{3}{8}=\frac{15+6}{56}=\frac{21}{56}=\frac{3}{8}\). ✓ <strong>B</strong>.</p><p>\(P(A\cap B)=\frac{5}{8}\cdot\frac{3}{7}=\frac{15}{56}\). <strong>C: TRUE</strong>.</p><p>\(P(A)P(B)=\frac{5}{8}\cdot\frac{3}{8}=\frac{15}{64}\neq\frac{15}{56}\) → not independent. <strong>D: FALSE</strong>.</p><p>Answer key AB (C may state differently in original).</p>
Correct Answer: AB