The “Boy or Girl paradox”

[BigT]

@Aspidistra

Actually, to solve this particular problem, you can dispense with an ordering altogether if you want. The only core fact you need to know is how much of your population falls into each group.

1/4 of all families have 2 boys
1/4 of all families have 2 girls
1/2 of all families have one of each

Now, like you say the reason that the population is structured like this is because the families were generated by random independent pickings. But for the purpose of this problem, we don’t actually have to care. We could say “that’s what the proportions look like, because logic” or “that’s what the populations look like because I went out and surveyed 1000 families” - either way, as long as you accept the validity of the proportions above, you can do the rest of the problem.

The question I was being asked was specifically the “why” part that you admit is because of the two distinct pickings.

It’s not like you actually did go out and do a survey, because, as stated upthread, the chances are not actually 50/50. You derived it from the assumptions.

[Gorilla]

@Thudlow_Boink

As septimus said earlier, just because you can break it down into mutually exclusive outcomes doesn’t mean you can automatically assume that all of them are equally likely. You could stipulate, as part of the conditions of the problem, that they are, but that would be a constraint.

The issue is not how likely each individual method of selection is (of two children). What matters is the biases of each parent(s) in terms of what combination of children they want (boy-boy, boy-girl or girl-girl). It doesn’t matter how they arrive at their chosen combinations. That is what makes the adoption scenario fundamentally different from the biological one.

The biological assumption (ignoring unusual cases like twins) is actually a very good stance to take purely for mathematical purposes. This is because it is exactly identical to a coin-toss probability problem involving two coins. The boy-girl paradox is just a more interesting way of presenting it. It is merely coincidental that, in the real world, the assumptions are pretty safe to make because they correspond with the large majority of actual cases.

@Gorilla

Yes, it’s basically the same thing. I should have made it clear that the answer is 1/2 for a very different reason, i.e. not simply because the chances of having a boy as opposed to a girl is 1/2.

I’d like to make a slight correction. While it is true that the calculations are very different, they nevertheless necessarily amount to the same thing. (Though, this is only true with the normal assumptions which makes the problem identical to a coin-toss situation). In other words, the two questions:

“A man with two kids was seen outside with his son. What’s the probability that both his children are boys?”

and

“A man with two kids was seen outside with his son. What’s the probability that the very specific child at home is a boy?”

Give exactly the same results. In other words, once you have already seen one kid, you might as well just focus on the other kid alone and ask “what’s the probability that that kid is a boy?” If you expand the problem to a scenario involving three children, you will still get the same identical outcomes:

“A man has three children. One day, he was seen with his son. What’s the probability that all three of his children are boys?”

The probability of him having being seen with a boy instead of a girl is 1 if from a BBB family. For BBG, GBB or BGB families, the probabilities are 2/3 each. For BGG, GGB or GBG families, the probabilities are 1/3 each. Therefore the probability of him being from a BBB family is 1/(1+2+1) = 1/4.

On the other hand, if you ignore the first child and simply ask “what is the probability that the two specific children at home are boys?” then the options are BB, BG, GG and GB. Thus the probability is still 1/4.

Therefore the normal intuitive answer (for the two-child case) of 1/2 is correct even though one may not be aware of all the considerations that go into it.