Question

One ticket is selected at random from $100$ tickets numbered $00,01,02,\cdots ,98,99$. If $x_1$ and $x_2$ denotes the sum and product of the digits on the tickets, then $P(x_1=9/x_2=0)$ is equal to

• A. $\frac{2}{19}$
• B. $\frac{19}{100}$
• C. $\frac{1}{50}$
• D. None of these

Answer 1

The correct option is A $\frac{2}{19}$

Let the number selected be $xy$. Then
$x+y=9=x_1,0\le x,y\le 9$ and
$xy=0=x_2\Rightarrow x=0,y=9$ or
$y=0,x=9$
$P(x_1=9/x_2=0)=\frac{P(x_1=9\cap x_2=0)}{P(x_2=0)}$
Now,
$P(x_2=0)=\frac{19}{100}$ and
$P(x_1=9\cap x_2=0)=\frac{2}{100}$
$\Rightarrow P(x_1=9/x_2=0)=\frac{2/100}{19/100}=\frac{2}{19}$