Which one?
A mythical city contains 100,000 married couples but no children. Each family wishes to "continue the male line", but they do not wish to over-populate. So, each family has one baby per annum until the arrival of the first boy. For example, if (at some future date) a family has five children, then it must be either that they are all girls, and another child is planned, or that there are four girls and one boy, and no more children are planned. Assume that children are equally likely to be born male or female.
Let p(t) be the percentage of children that are male at the end of year t. How is this percentage expected to evolve through time?
B = boys percent, and G = girls percent
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1 solution
At first, we think that people are producing children till they get a boy, so eventually the boy percent would be greater, but please note that if a family has 5 girls and the 6th boy, the percent of girls is much more.
Okay I know that the number of couples is really irritating you. So at first, lets just look at a smaller number.
Let us say that there are 100 couples. It is given that the probability of getting a girl is 50% So, 50 new-born are girls, and 50 are boys. So here the new ratio of girls to boys is 1:1 Thus, 50% are girls. Now, the parents of that 50 boys would not produce more children again.(sorry for my language) Lets talk about the parents of the 50 girls born. 25 of them will get a boy and stop. 25 will get a girl. Thus total number is now girls = 50 + 25, boys = 50 + 25. This will go on until each family gets a boy and the girl percent will be 50.
Similarly, for N number of couples, the percentage of Boys would always be equal to the percentage of Girls.