Question

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

Answer 1

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