Stumpokapow said:
Possible child configurations:
BB <-- could be this, other kid is boy
BG <-- could be this, other kid is girl
GB <-- could be this, other kid is girl
GG <-- could not be this because one of them is a boy.
Two in three chance it's a girl.
It's a Monty Hall style freakonomics counterintuitive thing.
Well done, this is absolutely correct.
If I had said the first is a boy, or the second is a boy, then the chance of the other being either a boy or a girl would have been 50/50. But saying that one is a boy makes the chance of the other being a girl 2/3.
It really is counter intuitive as most people with a basic understand of probabilities misapply the principals of independent events in this case.
For those of you that still have a hard time believing this, think of it this way. You are twice as likely to have one child of each gender then you are of having two boys (because there are twice as many combinations that produce that outcome).
Here's another way of putting it that gets people even more confused:
If I were to flip two identical pennies in secret, and show you that one of them was heads, there would be a 2/3 chance that the other was tails (for the same reasons as the problem above). However, if one the pennies had a mark on it, and you saw that mark when I revealed that it was heads, the chances of the other being tails is now 50%.
When presented with this, most people think it sounds ridiculous. How can the fact that there is a mark on one penny change the odds of what the other can be?
The answer of course, is that it can't. What has changed, in a subtle way, is the question. In the second example, the question is reduced to simply, "what was the result on the penny without the mark", which is clearly 50/50, completely independent of what happened to the one with the mark. In the first case, the question can be thought of "what are the chances he got 2 heads"?
In the second case, we have complete information about one of the coins, but none about the other, whereas in the first example we have some information about the combination of the coins.
Like I said, very unintuitive.