I was solving this problem:
In a world where everyone wants a girl child, each family continues having babies till they have a girl. What do you think will the boy to girl ratio be eventually? (Assuming probability of having a boy or a girl is the same)
The solution given was:
Suppose there are N couples. First time, N/2 girls and N/2 boys are born (ignoring aberrations). N/2 couples retire, and rest half try another child. Next time, N/4 couples give birth to N/4 girls and rest N/4 boys. Thus, even in second iteration, ratio is 1:1. It can now be seen that this ratio always remain same, no matter how many times people try to give birth to a favored gender.
My doubt is that will following be the case:
P(population will have more girls)= P(population will have equal number of boys and girls)= 1/2
Consider there are 16 couples
- 8 give birth to girls and hence stop. 8 give birth to boys, so they give another chance.
- 4 give birth to girls and hence stop. 4 give birth to boys, so they give another chance.
- 2 give birth to girls and hence stop. 2 give birth to boys, so they give another chance.
- 1 give birth to a girl and hence stop. 1 give birth to a boy, so they give another chance.
- Note till now there Number of boys = Number of girls
- Now probability that a single remaining couple give birth to a girl is 1/2. In that case they will stop and there will be one more girl than boys in the population. Probability that a single remaining couple give birth to a boy is 1/2, in which case they will give another chance in which they will again have a 1/2 probability of giving birth to boys, thus again balancing girl-boy ratio.
So am I correct with two facts:
Fact 1:
P(population will have more girls than girls)= P(population will have equal number of boys and girls)= 1/2
Fact 2:
P(population will have more boys than girls) = 0