[DougBTX]

How much do you need to know about the person to know which child is which?

I don't quite understand it, but apparently anything which can be used to distinguish the children will do.

Possibilities with two children:

    Gg, Bg, Gb, Bb

If one of them has a distinguishing mark, they have an apostrophe: (in jail, has red hair, or born first)

    G'g, B'g, G'b, B'b, Gg', Bg', Gb', Bb'

Then note that the marked one is a girl:

    G'g, G'b, Gg', Bg'

So, there is a 50% change that the children are a boy and a girl. Only if there is no way to distinguish them, do you get the 66% behaviour, where the set is:

    Gg, Bg, Gb

[seano]

No, that is not it. With G'g and Gg' you are repeating the same permutation in your set above!

It makes no difference if they have distinguishing marks or not. It matters if you are told that a particular child is a girl (50%) or if you are only told that at least one child is a girl (66%).

[DougBTX]

How can you be told information about a particular child if you have no way to distinguish them?

I partially understand your point about G'g vs Gg' now: if having a prime is the only way to distinguish the children, then G and g must be indistinguishable, so G = g.

[seano]

Think about it this way: I take two coins out of my pocket and hold them inside my hand so that neither of us has seen them. I show you the coin in my left hand and you see that it is tails, what are the odds of the coin in the other hand being heads? 50%. This is the chance of a head/tail combo in this case.

I now put the coins back into my pocket, shuffle them about, and again take them out inside my hands. This time I look inside both my hands, not letting you see, and tell you (truthfully) that at least one is tails. Given that information, you can deduce three mutually exclusive possibilities each of equal probability - both are tails, only the coin in my right hand is tails or only the coin in my left hand is tails. Hence we have the odds in this situation of 2/3 for a head/tail combo.

It is easy to see that the first situation is akin to knowing that a particular child is female, whilst the second is akin to knowing that at least one of the children is female. Also, in either case it does not matter if the coins are distinguishable - one could be a euro and the other a pound.