Boy Girl Paradox."WARNING" "Might seem complicated to some"

This will be a post regarding reasoning and logic.

Of delusions and truths.


The Boy And Girl Problem.


The Boy or Girl problem is a well-known example in the probability theory. Drop me and email or drop a comment if you would wish to know what is the probability theory.

- In a two-child family, the older child is a boy. What is the probability that the younger child is
a girl?

- A two-child family has at least one boy. What is the probability that it has a girl?

There are variations in the exact wording; often the second question confusingly asks about the "other child".
Although these questions appear to be of equal at first glance, at deeper investigation they turn out to be very different and to lead to different answers as well.

There are four possible combinations of children. Labelling boys B and girls G, and using the first letter to represent the older child, the sample space is

{BB, BG, GB, GG}.

These four possibilities are taken to be equally likely a priori. This prior follows from three assumptions: that the determination of the sex of each child is an independent event; that each child is either male or female and that each child has the same chance of being male as of being female. It is helpful to acknowledge these assumptions, not least because in the real world none are true. The ratio of boys to girls is not exactly 50:50, and amongst other factors the possibility of identical twins means that sex determination is not entirely independent.

The 1st question.

In a two-child family, the older child is a boy. What is the probability that the younger child is a girl?

When the older child is a boy, then the elements {GG} and {GB} of original sample space cannot be true, and must be deleted so that the problem reduces to:
{BB, BG}.

Since only one of the two possibilities in the new sample space, {BG}, includes a girl, the probability that the younger child is a girl is 1/2.

The 2nd question.

A two-child family has at least one boy. What is the probability that it has a girl?

In this question the order or age is not important. Therefore the set is:
{BG, GB, BB}
Therefore the probability is 2/3.

There is also a 3rd approach which is called the Bayesian approach which is exactly identical to the conclusion of the 2nd question but is derived is mathematical terms. Which is rather complicated. The Bayesian approach which consist of arithmatic theorems supports the 2nd question.

Conclusion.

The majority of people coming across this paradox for the first time will agree with the answer to the first question, but will consider the second as nonsense as it has to be of course the same as the first.
Two ways of explaining the error are as the following:
The second question does not assume anything about the age of the boy. He might be the older or he might be the younger sibling. Therefore the thought that there are only three possibilities
(2 boys {BB}, 2 girls {GG}, or a mix)

does not take into account that the last of these three is twice as likely as either of the first two, because it can be either {GB} or {BG}.

The chance that there are two boys is 1/4, the same as the chance that there are two girls. The chance that there is one boy and one girl (or one girl and one boy) consumes the remainder (1/2), therefore two boys are half as likely as a mixture.

Please... if there are any errors or flaws identified in this post, I wish to be notified of it.
Any disagreement, additional views are gladly welcome together with confusions that seems to be in need of clarity can seek me in comments or e-mail.

Have a nice day. Or night. Whatever the time is after you are done viewing this.