Thank you for this. I slugged through the whole original article and felt like I was being beat up with words.

However, I still fail to comprehend how the "at least one is a boy" quirk maths out to a 1 in 3 chance that his second child is also a boy.

Taken literally, it does. I understand that, in a set of data, GB is different from BG. But for the sake of our comparison, the order the children were born in doesn't matter. We're seeking if the other child is a boy, or not.

In my mind, the bit about it being the younger or older sibling is irrelevant information. We're comparing gender, not age. Regardless of if she is the younger sister, or the older sister, she's still his sister, and therefore not a boy.

I think the introduction of age is convoluting the issue, unnecessarily.

I now standby, ready to be proven wrong. I'd really like to wrap my head around this one, but I must insist that the age information is irrelevant.

Probability is much less meaningful when you're not talking about large groups of repeated trials.

In the context of the article, imagine that you didn't do this just one time, but that you asked 100 fathers about the compostion of their children.

The question posed in the article is essentially, "Of those fathers who responded 'I have one son,' (which is likely 75 of the 100), how likely is it that they have another son, (which is likely 25 of that group of 75, or 1/3).

When the article talks about the father standing next to one of his children at random and the probability of another son being 1/2 at that point, it helps to imagine those same 100 fathers all standing next to their children. Of that group, you're not eliminating the fathers standing next to girls based on the way the situation is posed.

The English words used to describe each case make it much less clear which group of 100 people we're talking about.

Also, when we talk about one father and not a group of fathers, the 1/3 or 1/2 number is much less meaningful. This is where insurance companies make their money (ideally). It's impossible to predict whether a single person will die in a car accident over their lifetime, and any number is essentially a guess. But it's very easy to predict that, say, 1 in 50,000 people will.

> However, I still fail to comprehend how the "at least one is a boy" quirk maths out to a 1 in 3 chance that his second child is also a boy.

Can you program? If so, write a program which runs the following trial over and over:

1. Assign genders at random to two children

2. If at least one is a boy, increment tally T1

3. If both are boys, increment tally T2

You'll find that T2 / T1 approaches 1/3.

You could equally well distinguish between BG and GB based on birth weight. Age has nothing to do with it. If the order of the children doesn't matter, the odds of conceiving a BG combination are twice as large as conceiving either a BB or a GG combination. The distinction is made so all combinations carry the same weight, which makes it easier to illustrate the solution.

As another alternative, you could distinguish between BG and GB based on a non-accidental property, like the alphabetic ordering of their names (assuming each starting letter is equally likely and no two siblings have the same name, both of which are probably false in practice).