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boys and girls

Hi,

First of all I would like to say that I like this site very much. There are many interesting problems and it is a big fun to try to solve them.

Let me show you a simpler solution to the boys and girls problem where you don't need to sum up an infinite row.

Every single child is born to be boy or girl with equal probabilty (50 percent). Since children just don't disappear after they got borned what else would the result be?

It is just a number of samples of the same probability variable, isn't it?

BTW the following note in the solution
"... with perhaps a few more boys than girls because the sum isn't actually infinite..." is not true.

At any given time there can be more girls than boys, you can even have millions of girls without any boys for decades even though sooner or later the rate should be close to 1 : 1.

Best regards,

Levy

Levente Mészáros
Thursday, November 25, 2004

You're right.

It seems that the only effect on population distribution this has is that every couple has exactly one boy.

I was at first skeptical of your answer, but verified your work by doing the following:

1) Assume that every couple has three children.  Then the following sequences occur with equal probability:

BBB
BBG
BGB
BGG
GBB
GBG
GGB
GGG

Here, we have 12 girls and 12 boys.  It is reasonable to expect that the distribution would be similar for any other fixed number of children.

2) If we adopt the rule that a couple stops having kids once they have a boy, we get the following sequences with equal probability:

B
B
B
B
GB
GB
GGB
GGG

Here we have 7 girls and 7 boys.

For the case of the GGG, we would extend the sequence to four kids but would see the same pattern.

As you stated, adopting this behavior does not suggest that a population must have at least as many boys as girls (or vice versa) but rather depends on how the probability distribution is sampled.  That is, if four couples each have a boy the first time, then the population will have more boys than girls but if the four couples have the sequence GGB, then there will be more girls than boys.

Jared
Saturday, December 04, 2004

Except  that the problem as given says nothing about the probability of boy vs girl birth.  Therefore everyone who has answered so far is wrong.  The correct answer requires the same infinite sum, but rather than (0.5)^n, you must sum whatever the probability of having a girl is.  If the probability is greater than one-half, the infinite sum diverges, implying more girls than boys.  If the probability is less than half, the sum converges but not to a ratio of 1:1.

This is similar (in concept, not math) to the subway teaser: if trains going North arrive at 15-minute intervals, and trains going South do as well, Bob figures he'll catch whichever train shows up next and end up visiting his North and South girlfriends equally often.  But he actually ends up headed North 3 times as often.  Why? Because the arrival times are not symmetric  but his arrival time at the platform is uniformly random.  Since the South train arrives shortly after the North, it's far more likely for Bob to wait on the platform until North shows up than in the short time interval between North and South arrivals.

Carl Witthoft
Thursday, January 06, 2005

this problem is dumb since every couples chance of having a boy is the same individually. what if they all had boys the first time, or 80% had boys, and the other 20% had a boy the second time. there are many scenarios where the boys outnumber the girls since you have to think of each couple seperately.

matt
Thursday, January 20, 2005

If there is a basic genetic tendancy for re-inforcement (stastical memory) violating iid without a counter re-inforcement (negative feedback) the solution will be similar to the of the "polyhedral urn" with red and blue balls sensitive to initial conditions.

the reality in biology is that there is a "group" stabilizing mechanism some believe hormones to keep a slightly positive bias on the male birth ratio about 102:100 due to increased risk taking in males and shorter re-productive life space.  There was a large occurance of male children born after WWII.

Outside of the biological scope, in the sociological scope many cultures practice infanticide creating according to Amaryta Sen 100million missing women in the world.  Suadi for example has a 125:100 M:F sex ratio, india and China also have a strong male bias reflecting selective birthing or infanticide of girls.

nick gogerty
Thursday, February 17, 2005

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