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Boys and girls

Mathematically, right answer, wrong reason. Genetically, wrong answer.
(Here we assume that the probabilty of either gender is 50%)

The question:
"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?
"

MATHEMATICALLY:

The answer:
" about 1:1"

Your explanation:
"Pretty simple. Half the couples have boys first, and stop. The rest have a girl. Of those, half have a boy second, and so on.
So suppose there are N couples. There will be N boys. There will be an "infinite" sum of girls equal to N/2 + N/4 + N/8 + ... As any college math student knows, this sum adds up to N. Therefore, the proportion of boys to girls will be pretty close to 1:1, with perhaps a few more boys than girls because the sum isn't actually infinite. "

My explanation: the same excpect that:
"
...There will be an "infinite" sum of girls equal to N/4 + 2N/8 + 3N/16 + 4N/32... whose sum does add up to N....
"

This is why:

1/2 have a boy and stop: 0 girls
1/4 have a girl, then a boy: N/4 girls
1/8 have 2 girls, then a boy: 2*N/8 girls
1/16 have 3 girls, then a boy: 3*N/16 girls
1/32 have 4 girls, then a boy: 4*N/32 girls
...
Total: N boys and
1N    2N    3N    4N
--  +  --  +  --  +  -- +... = ~N
4        8      16    32

Therefore, same answer, but I believe I have the correct proof.


GENETICALLY:

Answer: more girls (possibly)!

As you may know, the sperm determines the gender. Therefore, in reality, there are some complexities due to the fact that fathers who tend to have girls will have more girls, which in turn will pass on the father's genes further. If the trait of producing only girls is genetic (I don't know), then, in such a society, there would be MORE GIRLS!
On the converse, fathers who will tend to produce more boys will have fewer offsprings to carry their genes.

Also, it is a fact that some fathers can only produce girls. What would happen in that society? Theoretically, that father would have an infinite number of girls, resulting in a ratio of infinite:1! In reality, such a man (assuming monogamy) would be able to father 20 girls or so. That is still enough to skew the proportions in one generation. I don't how future generations would be affected, because I don't know if that is a genetic trait. If it is, the number of girls would increase exponentially, as more girls that carry that gene become mothers and give birth to sons who cannot in turn produce sons. Finally, the grouth would taper off, balancing the supply of boys from other families who do not have that gene.


Davide

Davide Andrea
Tuesday, December 11, 2001

Wait a second!
Assuming that each time a baby is born the probability it will be a boy is 50% and the probability that it will be a girl is 50%:

The ratio is 1:1, but (I think) I have a much simpler proof than yours:

If there are N children born, since the probability that each child be of a particular gender is 1/2, there will probably be N/2 boys and N/2 girls.

Do the *reasons* they decide to have children matter?

(Note: Many people here are far better than I am mathematically. If I'm wrong, please tell me, gently.)

Dan
Sunday, December 16, 2001

Dan,

You're very correct.  Quite a clever way to think about it, I think.

Chad

Chad Hulbert
Monday, December 17, 2001

Wow, Dan, you're right.  This is a trick question, and I think you're the only one who realized it, including the original asker!

Paul Brinkley
Tuesday, December 18, 2001

Thank you.

I was *sure* that I was missing something, becasue it seemed so simple...

Dan
Thursday, December 27, 2001

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