## Saturday, October 13, 2007

### Keep trying until you get a male: Still equal sex split.

Opus the Penguin wrote:
> 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'm not sure how to set that one up so I can do some figurin'.

Let's suppose that p(male) is .5 (it's not quite).

Here's the setup, assuming the couple stops at 10 children.
Half the time (.50) they get a male first, then stop.
The other half they get a female first, and continue.
Half of that time (.25) they get a male second then stop.
The other half, they get a female second and continue.
Half of that time (.125) they get a male third then stop.
(etc.)
By the time we get to 10 children under this rule, we have exhausted 99.9512% of the possible alternatives, and have exhausted at least one parent. The average number of males is .999 across families. The average number of females is .994.

p males females
0.50000 1 0
0.25000 1 1
0.12500 1 2
0.06250 1 3
0.03125 1 4
0.01563 1 5
0.00781 1 6
0.00391 1 7
0.00195 1 8
0.00098 1 9
0.00049 0 10

Average number of males females
0.999 0.994

p(male)
0.5

Now, what happens is the probability of a male is, say, 90%? There are the same options, but their probability changes and males are much more likely.

p males females
0.90000 1 0
0.09000 1 1
0.00900 1 2
0.00090 1 3
0.00009 1 4
0.00001 1 5
0.00000 1 6
0.00000 1 7
0.00000 1 8
0.00000 1 9
0.00000 0 10

Average number of males females
1.000 0.111

p(male)
0.9