Following the puzzling tradition, I thought I’d post a puzzle. A real probability puzzle. I have constructed it just now, so you can’t google for the answers.
But first the background. I read here that ” Short women [are] more successful with men”

“The average height for a British ?oman is 1.62 metres (5 feet, 4 inches). But those who were between 1.51 and 1.58 metres were most likely to be married and to have children by the age of 42. This relationship held true even after accounting for social class.
The study also found that women prefer men who are taller than average. A man of 1.83 metres (6 feet exactly) was more likely to have a partner and children than a man standing at the average height of 1.77 metres (5 feet, 10 inches).”

Er.. Did the study really find that women “prefer men taller than average”? Or did it just find that they prefer men taller than they are?
Anyway, here is the problem:
Suppose that human beings come in 5 sizes: 4′ , 4′ 6″, 5′, 5’6″, 6′. Assume that initially, the height distribution is even among both men and women (i.e., 1/5th of people are 4 feet tall, 1/5th 4 1/2 feet and so on).
Each person meets 10 people of the opposite sex, at random in his or her lifetime. Of these 10 people men will not consider any woman taller than they are, and women will not consider any man shorter than they. Among those that remain, there is a 10% chance that the man and woman will hit it off and marry.
Once they marry, every couple has 2 children. One of them(boy or girl – doesn’t matter) will have the husband’s height and the other the wife’s height.
Problem 1 Find out the height distribution among men and women after 5 generations.
Problem 2 Is there an ‘equilibrium’ height distribution, and if so, after how many generations will it be achieved?
Advanced Problem: We have assumed that men are neutral to a woman’s height as long as the latter are shorter( i.e., a 5 1/2 feet man is equally likely to marry a 5 feet woman as he is likely to marry a 4 feet woman.) Suppose that this is not true. Suppose that there men are biased in favour of women one size shorter, and women biased in favour of men one size taller. How much of this bias is required to eliminate the tendency towards shortening of women and lengthening of men?