Question

Let $X$ and $Y$ be two events such that $P(X \mid Y)=\frac{1}{2}, P(Y \mid$ $X)=\frac{1}{3}$ and $P(X \cap Y)=\frac{1}{6}$. Which of the following is(are) correct?

• A. $P(X \cup Y)=\frac{2}{3}$
• B. $X$ and $Y$ are independent.
• C. $X$ and $Y$ are not independent.
• D. $P\left(X^{C} \cap Y\right)=\frac{1}{3}$

Answer 1

Answer: (A), (B)

$P\left(\frac{X}{Y}\right)=\frac{P(X \cap Y)}{P(Y)}=\frac{1}{2} $
and $ \frac{P(X \cap Y)}{P(X)}=\frac{1}{3} . $
$ P ( X \cap Y )=\frac{1}{6} \Rightarrow P ( Y )=\frac{1}{3} $
and $P ( X )=\frac{1}{2} .$
Clearly, $X$ and $Y$ are independent. Also,
$P(X \cup Y)=\frac{1}{2}+\frac{1}{3}-\frac{1}{6}=\frac{2}{3}$