jmc

The positive multiples of 45 are $$45,90,135,\dots ,990=1\cdot 45,2\cdot 45,3\cdot 45,\dots ,22\cdot 45.$$ There are 22 multiples on this list. Every positive multiple of 45 less than 1000 is either a two-digit integer or a three-digit integer. Out of the $99-10+1=90$ two-digit integers, $45$ and $90$ are multiples of 45. Therefore, the probability that the selected multiple of 45 has two digits is $2/22=\boxed{\frac{1}{11}}$.