Question

Consider two independent random variables X and Y with identical distributions. The variables X and Y take values 0, 1 and 2 with probability 1/2, 1/4 and 1/4 respectively. What is the conditional probability P(X + Y = 2/X - y = 0) ?

• A. 0
• B. 1/16
• C. 1/6
• D. 1

Answer 1

The correct option is C 1/6

$P\left(\frac{X+Y=2}{X-Y=0}\right)=\frac{P\left[\right(X+Y=2)\cap (X-Y=0\left)\right]}{P(X-Y=0)}$

$=\frac{P(X=1,Y=1)}{P(X-Y=0)}$

$=\frac{P(x=1).P(y=1)}{P(x=y)}$

x	y	P
0	0	$\frac{1}{2}$
1	1	$\frac{1}{4}$
2	2	$\frac{1}{4}$

$=\frac{\frac{1}{4}\times \frac{1}{4}}{\frac{1}{2}\times \frac{1}{2}+\frac{1}{4}\times \frac{1}{4}+\frac{1}{4}\times \frac{1}{4}}=\frac{1}{6}$