First thing to note is that in the 2nd draw, we will always use the same urn as the 1st draw, instead of independently select the urn again.

Next by Law of Total Probability,

The crucial fact here is that maybe no longer equal to ; and it require you to apply the Baye's Theorem to calculate them.

(That's why this question is put in this section.)

You should be able to figure out the remaining part - you already know