I'm posting one puzzle, riddle, math, or statistical problem a day. Try to answer each one and post your answers in the comments section. I'll post the answer the next day. Even if you have the same answer as someone else, feel free to put up your answer, too!
Thursday, August 07, 2008
Boys or Girls
In a country in which people only want boys, every family continues to have children until they have a boy. if they have a girl, they have another child. if they have a boy, they stop. what is the proportion of boys to girls in the country?
I say it is 50/50 because no matter how many babies you have, the likelihood of having a boy is 50%, so no matter when you start or stop having children, the ratio is 1:1
Anon, your logic is right on. For a more mathematically rigorous explanation, the key here is to consider all the different cases and their respective probabilities. The family will have n children, some number of girls before then a boy, and P(n) is the probability of that case. The expected value for any probability density function is the sum over all n of n*P(n).
P(1) = 1/2 P(2) = 1/4 P(3) = 1/8 P(4) = 1/16
So, n-bar (expected value for n) is:
1/2+2/4+3/8+4/16+... = 2
That means the average value for number of children is 2, one of which is a boy. So yes, it's still 50/50. Crazy how math works and stuff...
i do feel the urge to chime in that the Y sperm is lighter than the X sperm since it's missing 800 or so genes and therefore the chance of having a male child concieved is increased but the chance of a male child miscarraige (cause the mother's body recognizes a flaw in the child due to missing genes) is also increased. More male babies are born that female, but more females live to a ripe old age than males because of superior genetics. (please don't take this as sexist)
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I say it is 50/50 because no matter how many babies you have, the likelihood of having a boy is 50%, so no matter when you start or stop having children, the ratio is 1:1
ReplyDeleteAnon, your logic is right on. For a more mathematically rigorous explanation, the key here is to consider all the different cases and their respective probabilities. The family will have n children, some number of girls before then a boy, and P(n) is the probability of that case. The expected value for any probability density function is the sum over all n of n*P(n).
ReplyDeleteP(1) = 1/2
P(2) = 1/4
P(3) = 1/8
P(4) = 1/16
So, n-bar (expected value for n) is:
1/2+2/4+3/8+4/16+... = 2
That means the average value for number of children is 2, one of which is a boy. So yes, it's still 50/50. Crazy how math works and stuff...
As always Abe, a great explanation.
ReplyDeletei do feel the urge to chime in that the Y sperm is lighter than the X sperm since it's missing 800 or so genes and therefore the chance of having a male child concieved is increased but the chance of a male child miscarraige (cause the mother's body recognizes a flaw in the child due to missing genes) is also increased. More male babies are born that female, but more females live to a ripe old age than males because of superior genetics. (please don't take this as sexist)
ReplyDelete