Similar puzzles with different answers

[bob j]

I have a question about two puzzles that seem similar to me, but I have different answers to them.

One is the following:
You have a hat in which there are three pancakes: One is golden on both sides, one is brown on both sides, and one is golden on one side and brown on the other. You withdraw one pancake, look at one side, and see that it is brown. What is the probability that the other side is brown?

The answer is 2/3 and uses the fact that you had twice as much of probability of choosing the pancake with double brown.

The other problem is this:
Ms. X has two children. You see her walking with one of her children and that child is a boy. What is the probability that both children are boys?
I found this on a book and the answer was 1/2.

Are these different problems? They look the same to me

Thanks

 

[LeonhardEuler]

These are different problems. In one case, you know how many golden and brown pancake sides there are. Finding a brown one gives you information about which pancake you picked. In the other case, you have no idea what the genders of the two children are, so the gender of one gives no information about the gender of the other. If you knew there was one boy and one girl, or two boys, you would be able to say the gender of the one not present with certainty, but you don't have this information, so information about one says nothing about the other.

 

[bob j]

could you possibly tell me how to derive that mathematically? It is not clear to me why knowing one side of the pancake is brown is not equivalent to knowing that one child is a boy.

Thanks a lot

 

[LeonhardEuler]

For the pancake problem, I'm equally likely to pick any side of any pancake. I picked a brown side. There are three brown sides, two of them on the all brown pancake, one on the half brown one. The odds that I picked the all brown pancake then must be 2/3.

For the other case, knowing one kid is a boy, that gives me no information about the other one since I don't know the total numbers.