A probability word problem, it's not a riddle, no trick answer. You have 3 identical boxes. One contains 2 gold marbles, one 2 silver marbles, one 1 of each. You can't look in the boxes. You pick one of the boxes at random. You reach in and pull out a gold marble. What are the odds the second marble you pull out of the box will be gold?
On your initial draw, you have pulled one of three gold marbles, call them b1m1 (box 1 marble 1), b1m2, and b3m1. You know you're drawn one of them, but you don't know which one. Of the three possibilities, in two cases the second marble in the box is also gold, so it's 2/3. This is somewhat related to the Monty Hall problem.
It's counter intuitive. The way to think about it is this. Two of the three boxes contain two of the same colour marbles. So when you select the first box at random, there is a 2/3rds chance it contains two like coloured marbles. Looking at one marble doesn't change that, whether it is gold or silver, 2/3rds of the time the second marble will be the same as the first. Since 2/3rds of the time the box you randomly selected contains two like coloured marbles.
Forgive my flawed brain, but I have to disagree with Bertrand. I grant that prior to picking a box or marble the chances are 2/3 that you will pull 2 marbles of the same colour from the one box. But only a 1/3 chance that is will be 2 gold or 2 silver. By pulling a gold out, you have eliminated the box with the 2 silvers. So you are choosing from either a box with a remaining gold or a silver marble in it. That is only 50% at that point in time. Yet I can see, if you read the question differently, that the 2/3 option is there. I have trouble with Schrodinger's Cat also.
I don't know that I can explain it, but I think I get it now.
It's related not just to which box is picked (1 in 3) but to the FACT that you drew a gold marble as your first draw.
Think of the chances of drawing gold from either of the two boxes that have gold and thus the chances that you drew gold AND got the box with one gold and one silver... since you drew gold as your first draw, you are more likely to have picked up the box with two gold in it.
The part I can't explain is how the math rules that this means the chancesa re 2 in three. I probably could if I spent more ergs on this... but I am feeling too lazy for that.