Question

Suppose there is a population with the tradition that the women bear children until they have one boy. What would be the ratio of boys to girls in this population?

Suppose that the probability to give birth to a boy is the same as to give birth to a girl. Also, the women in this population keep bearing children until they have exactly one boy.

Answer 1

`50:50`

Explanation:

There are two ways of explaining:

1. (Logic)

By introducing the rule that women keep bearing children until they have a boy, the number of children is effected (`n`) and not the probability (`p`) and therefore not the ratio of the gender of the children. The probability is still `1/2` and therefore the ratio is still `50:50`.

2 (Mathematic :) )
To determinate the number of boys or girls in one family, we need to find the expacted value of the number of childrens within one family.
`E(x)=1*P(1)+2*P(2)+...+n*P(n)`
`E(x)=1*1/2+2*1/4+...+n*1/(2^n)`
`E(x)=lim_(n->oo)(sum_(i=0)^(n)(i/2^i))`

`sum_(i=0)^(n)(i/2^i)\stackrel{!}{=}(2^(n+1)-n-2)/2^n`
To prove this by induction, we need to show that this is true for `n=0` and that it is true for `n+1`

Base case
`sum_(i=0)^(0)(i/2^i)\stackrel{!}{=}(2^(0+1)-0-2)/2^0`
`0\stackrel{!}{=}(2-2)/1`
`0=0`

Inductive step

`sum_(i=0)^(n+1)(i/2^i)\stackrel{!}{=}(2^(n+1+1)-(n+1)-2)/2^(n+1)`
`(n+1)/2^(n+1)+sum_(i=0)^(n)(i/2^i)\stackrel{!}{=}(2^(n+2)-n-3)/2^(n+1)`
Sub in `sum_(i=0)^(n)(i/2^i)\stackrel{!}{=}(2^(n+1)-n-2)/2^n`
`(n+1)/2^(n+1)+(2^(n+1)-n-2)/2^n\stackrel{!}{=}(2^(n+2)-n-3)/2^(n+1)| * 2^(n+1)`
`(n+1)+2 * (2^(n+1)-n-2)\stackrel{!}{=}2^(n+2)-n-3`
`n+1+2^(n+2)-2n-4\stackrel{!}{=}2^(n+2)-n-3`
`2^(n+2)-n-3\stackrel{!}{=}2^(n+2)-n-3`

Q.E.D.

Therefore,
`E(x)=lim_(n->oo)((2^(n+1)-n-2)/2^n)`
`E(x)=lim_(n->oo)(2+(-n-2)/2^n)`
`E(x)=2-lim_(n->oo)((color(blue)(n+2))/color(red)(2^n))`

Since `lim_(n->oo)(color(red)(2^n)>color(blue)(n+2))`
`lim_(n->oo)((color(blue)(n+2))/color(red)(2^n))=0`

`E(x)=2-0=2`

Knowing that the ratio is
`color(red)(E(x)-1):color(blue)(1)`
`(color(red)(red)=girls, color(blue)(blue)=boys)`
We get the answer
`(2-1):1`
`<=>1:1`

The ratio of boys to girls is `1:1`