Answer can not be correct if statement "each family continues having babies till they have a boy" is correct


#1

If the statement “each family continues having babies till they have a boy” is true the ratio of boys to girls must be less than 0.5 if there is a 0.5 probability that a child born to a family is a boy.
Surely, if a probability tree is used the workings are;
Of each 1 family ;
=> 0.5 families have a Boy and STOP having children
=> 0.5 families have a Girl, continue having children and of each of these families;
=>> 0.25 families have a Boy and STOP having children
=>> 0.25 families have a Girl, continue having children and of each of these families;
=>>> etc, etc

So, if if the maximum number of boys a family can have is 1 and the maximum number of girls a family can have is infinite (as they do not stop until they have a boy) how can the ration of boys to girls ever be 1:1?
In fact, using infinity as the maximum number of girls per family, the result must tend towards a ratio of 0:1 !!

Can anyone explain why the logic above is incorrect?


#2

I think you answered your own question! Let’s consider the number of families to be N. Now, as you said, in the first round, 50% families have boys => # of girls = N/2. In the second round, again 50% of the families who had girls will have boys => # of girls = N/4. As we it’s mentioned, they have kids until they have a boy, we know # of boys will be N. And # of girls is N/2 + N/4 + N/8 + … and so on. Thus the ratio is N/N(1/2 + 1/4 + … ) = 1/(1/2 + 1/4 + 1/8 + …). Now 1/2 + 1/4 + 1/8 + … ≈ 1 => Your ratio is 1/1 = 1.00


#3

The question doesnt say “until they have boys”, it says: “until they have the gender they expected”. Meaning, they have one of each, never stop at the first, cause all the families continue to have kids… so, since its 50/50, the proportion is 1. Its very easy question.