Answer:
Maximum 3 socks (minimum 2) to be taken out to have a pair of same color.
Explanation:
Well, it can be assumed that for example there are 4n black and 5n brown socks in a drawer (which satisfies ratio of 4/5). If first sock is taken out the probability of this sock being black is 4/9 (4n/(4n+5n)) or being brown is 5/9 (5n/(4n+5n)). If the first sock is black, the probability of second attempt being black is (4n-1)/(9n-1), but being brown is (5n)/(9n-1). f the first sock is brown, the probability of second attempt being brown is (5n-1)/(9n-1), but being black is (4n)/(9n-1). In either case, there is a probability of taking out different colors of socks in two attempts (taking black first then brown or vice-versa). So maximum 3 (minimum 2) attempts required to make sure you have a pair of the same color of socks.