## February 24, 2015

### World with All Girls !!

In a world where everyone wants a girl child, each family continues having babies till they have a girl.
What do you think will the boy to girl ratio be eventually?
(Assuming probability of having a boy or a girl is the same)

Anonymous said...

1boy:2girls

Jerry Critter said...

Boy-girl ratio with still be 1:1.

Josh D. said...

I agree that it will be 1:1.

The average number of girls in each family is stated in the puzzle to be equal to 1 (since each family has children until they have one girl).

The average number of boys in each family is equal to sum of the number of boys in a family times the probability that a family has that many boys. Mathematically, this is:
SUM (from n=1 to infinity) of: (n-1) * .5^(n)
Writing out these terms, you will see they are:
0 + 0.5 + 0.25 + 0.125.... which sums up to 1.

Thus, the total ratio is 1:1.

Anonymous said...

2:1

If a Geometric sequence is: a, ar, ar^2, ar^3, ar^4, ar^5...

Then the equation for the Sum of a Geometric sequence that converges is S = a/ (1-r)

so: S = 1/ 1-1/2, which is the equation for the amount of girls per boy. S = 2

Jerry Critter said...

Chances of having a boy or a girl are even. So, let's start with 100 babies. 50 girls and 50 boys. The people with girls stop having babies.

So, 50 more babies are born, 25 girls and 25 boys. For a total of 75 girls and 75 boys. ( a 1:1 ratio)

People with boys, now 25, have babies again, and again there will be an equal,number of boys and girls born, continuing the ratio of 1:1.

1:1 is th correct answer, not 2:1.

Josh D. said...

I disagree. It should be 1:1. I explained the math earlier - your equation does not apply here.

Intuitively, think of it this way - The probability that any given woman has a boy or girl is equal.
Start with 256 pregnant women. They will have 128 boys and 128 girls among them.

The 128 women that had girls will stop, and the 128 that had boys will get pregnant again. Among these, there will be 64 more girls and 64 more boys (totals of 192-192....1:1).

The 64 women that had girls on the 2nd try will stop, and the 64 that now have two boys will get pregnant again. This will yield 32 more boys and 32 more girls (totals of 224-224....1:1).

It goes on like this infinitely.