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jll6m
05-05-2010, 04:05 PM
Test question:

Assume that it is equally likely for a pregnancy to deliver a baby boy or a baby girl. Suppose that for a particular large community of people, every family continues to have children until they have 2 boys, then they stop having children. After 15 generations of families, what is the ratio of males to females?

Teacher Answer:

Sex ratio does not depend on family size. Girls and boys are equally likely. Therefore:

M:F = 1:1

Question:

Is the teacher's answer correct? Why? He has been wrong before and this is not for a probability class so it is possible.

What I don't understand is that for the case where 2 boys are immediately born, 0 girls are born, and then they stop. So boys can never get above two per family. However, the family may have 100 girls before having 2 boys. Just a little confused.

BGM
05-06-2010, 03:51 AM
Is that each generations are independent?
If yes, in each generation,
the number of boy = 2
the number of girl, N ~ NegativeBinomial(2, 1/2)
Pr{N = n} = (n + 2 - 1)C(2 - 1)*(1 - 1/2)^2*(1/2)^n = (n+1)/2^(n+2), n = 0, 1, 2, ...
Ratio of boy to girl = 2 : N which is random

After 15 generations,
number of boys = 2*15 = 30
Note the sum of i.i.d. negative binomial random variables is still negative binomial
distributed. N1 + N2 + ... + N15 ~ NegativeBinomial(2*15=30, 1/2)

So it depends on what ratio you want.
30 : (N1 + N2 + ... + N15) is a random ratio
If you want the "expected ratio",
E[N1 + N2 + ... + N15] = 15E[N] = 15*2*(1/2)/(1 - 1/2) = 30
So the "expected ratio" = 30 : 30 = 1 : 1

SmoothJohn
05-06-2010, 10:45 AM
This is a nice exercise to simulate. I used Excel to model the situation with random assignment for each birth. I interpreted 1 as a boy and 0 as a girl.

1. Fill a row of 30 cells with =round(rnd(),0)
2. A2==IF(SUM(A$1:A1)=2,"",ROUND(RAND(),0))
3. Fill the A2 formula to the right across your 30 cells and down 10 cell or so.
4. At the bottom of each column in your array, calculate the number of girls in each family with Sum. (Of course the number of boys is always 2).
5. Average the number of girls in the 30 families.

I ran it 15 times, and had an average of 2.04 girls per family, which suggests that your teacher is probably right.

Of course, it is another matter to prove the conjecture. But I think it is worthwhile to first make a numerical model just to get a feel for the problem.

jll6m
05-06-2010, 12:21 PM
Thanks for the replies. Makes enough sense.

Berley
05-06-2010, 12:25 PM
Setting aside formulas for a moment, look at the question carefully. The parameters given include a "large community" and 15 generations. That's a lot of births. That's key to understanding this question.

Each birth is an independent event. Each child born (and therefore each person added to the community) has a 50% chance of being male and a 50% chance of being female. It doesn't matter how many girls have been born into the family already -- the next sibiling's odds are still 50-50.

The law of large numbers tells you that you should expect the observed ratio to approach the expected ratio. Your expected ratio is 1:1, so over a long period of time in a large community, you should expect the genders of occupants to be about 1:1 as well.

Remember, human families are limited in the number of children they can produce. You can't have 100 girls before getting 2 boys. 10 girls and 2 boys -- entirely possible, but starting to get unlikely.