[punter11235]

I suck at those but here is my try :

You could draw 2 red balls in (15 2) = 105 ways (every pair with same probabiilty).
2 red balls from first bag in (10 2) = 45 ways

So its 45/105 = 9/21.

Explanation :
I think its like that : there is 190 possibilites of drawing (including green balls). We are only interested in draws which has 2 red balls , they are 105 of them (also there are 75 draws with one green and on red and 10 with both green).
So now we calculate probability of drawing 2 red balls from 1st bag assuming one OUT OF THOSE 105 draws happened because we know drawing 2 red balls already happened (bayes theorem).
So its 45/105.

EDIT : Sry for my english I dont have time to correct it now I hope reasoning is clear; would love to have somebody smart to comment on it.