In a club, a member is either an Indian or a non-Indian who is either a man or a woman. One-third of them are women, two-thirds of them are Indian and three-eighths of the non-Indians are women. What is the probability that a man picked at random is a non-Indian?

In this, there are two kinds of separations. Men and women, Indian and Non-Indian. It is given that 2/3 of all are Indians. From this, we deduce that 1/3 are Non-Indians. Of these 1/3 Non-Indians, 3/8 are women. Now we can derive the Non-Indian man, it is (1/3)*(1-(3/8))=5/24. 5/24 of all are Non-Indian men. It is given that 1/3 of all are women, so 2/3 are men. Of 2/3 men, 5/24 are Non-Indians, so Indian men are 11/24. The probability of a picked man being Non-Indian is (5/24)/(2/3). That is 5/16.