incremental

It confirms with you, no trust required.

"Under your inference, the man wouldnt have mentioned anything unless he had at least one male child. (in which case you can say the GG scenario is gone, but GB BG and BB are equally probable)"

No. I'm just assuming he has two children, and randomly mentions something about one of them. The GG scenario is only eliminated after he makes his statement, because we then know he has at least one boy.

So you say: "it is clear why BB, BG, GB, and GG have equal probability in the unrestricted case"

The restricted case is just the unrestricted case + one additional bit of information, that is, you're told that GG is not an option. This eliminates GG from the unrestricted case, but says nothing more about the probabilities of the other options. So the probabilities stay equal, although they now equal 1/3 each (if you eliminate options, the remaining options all become more likely).

What you're missing is this: The statement "the older child is a boy" has more information than the statement "one of the children is a boy". The first statement allows you to eliminate two options (GB and GG), while the second statement only allows you to eliminate one option (GG).

The "older" part is not fundamental to the problem. Equally, the statement "the taller child is a boy" has more information than "one of the children is a boy". The problem with this is that probabilities for height are not so friendly like the 50/50 probabilities for birth order (e.g., boys are likely to be taller than girls, older children are taller than younger, etc), which introduces unnecessary complexities to a logic problem. So that's why birth order is used for these types of puzzles.

Imagine a similar problem but with red and blue poker chips. Say, for example, that I have a bag and I pull out two chips, one at a time.

In this problem, would you still try to distinguish the two identical red poker chips using your logic?

Since the constraint simply says that at least one child is a boy, we don't need to distinguish between two types of BB.

This is a curious misconception, by the way. The typical failure mode I see on these questions is people having difficulty accepting BG and GB as different possibilities.

For a randomly selected family with two children, there are four possible boy/girl combinations: B B, B G, G B, G G

In the first case we are told that the older child is a boy. This leaves only two cases: B B, B G Therefore, there is a 50% chance the second child is a boy.

In the second case, we are told only that [at least] one child is a boy. This leaves three possibilities: B B, B G, G B Therefore, the probability that both children are boys is 1/3.

Enumerating possible states of the world like this is the fundamental insight you need to have to be able to understand these types of problems - but it does take a while to get used to!

Not really a product, though, so that doesn't count as a good example. Obviously a major discovery with high impact, though.

Print edition date.

The funny thing is that he is clearly thinking like a computer programmer in his first sentence. I wonder if naive users would generally have better results on first contact with Siri, compared to technical users who try to over-structure their commands?

Planck's constant is not the smallest unit of distance. It's not a unit of distance at all.

You're thinking of the Planck length. It is indeed proposed that the Planck length is the smallest meaningful length, but this is by no means proved.