Probability
Conditional Probability and Independence
Grade 12
Question:
<p>\(P(X|Y)=\dfrac{1}{2}\), \(P(Y|X)=\dfrac{1}{3}\), \(P(X\cap Y)=\dfrac{1}{6}\). Which of the following are correct? <em>[JEE Advanced 2017]</em></p>
P(X\cupY) = 2/3
X and Y are independent
X and Y are not independent
P(X̄\capY) = 1/3
Step-by-Step Solution
Key Concept: From P(X|Y) = 1/2: P(Y) = P(X\capY)/P(X|Y) = (1/6)/(1/2) = 1/3. From P(Y|X) = 1/3: P(X) = 1/2.
<p>$P(Y)=\frac{P(X\cap Y)}{P(X|Y)}=\frac{1/6}{1/2}=\frac{1}{3}$. $P(X)=\frac{P(X\cap Y)}{P(Y|X)}=\frac{1/6}{1/3}=\frac{1}{2}$.</p><p>$P(X\cup Y)=\frac{1}{2}+\frac{1}{3}-\frac{1}{6}=\frac{2}{3}$. <strong>A ✓</strong>.</p><p>$P(X)P(Y)=\frac{1}{2}\cdot\frac{1}{3}=\frac{1}{6}=P(X\cap Y)$. So X and Y ARE independent. <strong>B ✓, C ✗</strong>.</p><p>$P(\bar X\cap Y)=P(Y)-P(X\cap Y)=\frac{1}{3}-\frac{1}{6}=\frac{1}{6}\neq\frac{1}{3}$. <strong>D ✗</strong>.</p>
Correct Answer: AB