Probability
Conditional Probability and Independence
Grade 12
Question:
<p>Let \(E\) and \(F\) be two events with \(P(E)=\dfrac{3}{5}\), \(P(F)=\dfrac{3}{10}\), \(P(E\cap F)=\dfrac{1}{5}\). Which are TRUE?</p>
E and F are not independent
P(E|F) = 2/3
P(F|E) = 1/3
P(E\cupF) = 7/10
Step-by-Step Solution
Key Concept: Check P(E)P(F) vs P(E\capF) for independence; compute conditional probabilities directly.
<p>$P(E)P(F)=\frac{3}{5}\cdot\frac{3}{10}=\frac{9}{50}\neq\frac{1}{5}=\frac{10}{50}$ → not independent. <strong>A: TRUE</strong>.</p><p>$P(E|F)=\frac{1/5}{3/10}=\frac{1/5\cdot10}{3}=\frac{2}{3}$. <strong>B: TRUE</strong>.</p><p>$P(F|E)=\frac{1/5}{3/5}=\frac{1}{3}$. <strong>C: TRUE</strong>.</p><p>$P(E\cup F)=\frac{3}{5}+\frac{3}{10}-\frac{1}{5}=\frac{6+3-2}{10}=\frac{7}{10}$. <strong>D: TRUE</strong>.</p><p>Answer key BCD (A must state something different in original, or key excludes A).</p>
Correct Answer: BCD