Re: Probability questions- plz help

Posted by Amit Goyal on May 31, 2013; 3:38am
URL: http://discussion-forum.276.s1.nabble.com/Probability-questions-plz-help-tp7581507p7581519.html

If the probability of having a male child is 0.5 and the probability of having a female child is also .
suppose a family stops when there are 2 male children or when they have 4 kids in  all.
what is the sex ratio of a population with such a family planning?

We can do it in multiple ways:
One is by quick observation. First child is equally likely to be a boy or a girl. So, half of the first borns' are boys and half for them are girls. Second child is again equally likely to be a boy or a girl. Again half of the second borns' are boys and half for them are girls. Now  only families with fewer than two boys so far will have the third child. However, even in this case, since the sex of the third child is independent of the first two, we have half of the third borns' are boys and half of them are girls. And similarly for the forth. Thus  1/2 of the population is male and half is female. Thus, the sex ratio is 1.
 
Alternatively, we can also do it by sample space method. Check that the sample space is given by,
S= {BB, BGB, GBB, GGBB, GBGB, BGGB, BGGG, GBGG, GGBG, GGGB, GGGG}
and the associated probabilities are:
Pr(BB) = 1/4
Pr(BGB) = Pr(GBB) = 1/8
Pr(GGBB) = Pr(GBGB) = Pr(BGGB) = 1/16
Pr(BGGG) = Pr(GBGG) = Pr(GGBG) = Pr(GGGB) = Pr(GGGG) = 1/16
Probability can also be interpreted as proportion of families. For example, Pr(BGB) is the probability that the family will have first boy child, second girl child and third boy child or the proportion of families with first boy child, second girl child and third boy child.
Male ratio =  Average number of male-children per family/Average number of children per family
              =  [2(1/4 +1/8 + 1/8 + 1/16 + 1/16 + 1/16) + 1(1/16 + 1/16 + 1/16 + 1/16) ] / [2(1/4) + 3(1/8 + 1/8)  + 4(1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16)]
              =  [2(11/16) + 1/4] / [2(1/4) + 3(1/4) + 4(1/2)] = [26/16] / [13/4] = 1/2
This implies proportion of females is also 1/2.
Thus, the sex ratio is 1.