Question

One ticket is selected at random from $100$ tickets numbered $00,01,02,\cdots 98,99$. If $X$ and $Y$ denote the sum and product of the digits on the tickets, then the value of $57P(X=9/Y=0)$ is:

Answer 1

$(Y=0)$ is $\{00,01,\cdots 09,10,20,\cdots ,90\}$
Also, $(X=9)\cap (Y=0)=\{09,90\}$
We have $P(Y=0)=\frac{19}{100}$ and $P\left[\right(X=9)\cap (Y=0\left)\right]=\frac{2}{100}$
Hence. $P(X=9/Y=0)=\frac{P\left[\right(X=9)\cap (Y=0\left)\right]}{P(Y=0)}=\frac{2}{19}$
$\therefore 57P(X=9/Y=0)=6$