Riddle

Weird problem for you math-y types. Say you have 2 boxes, each with money in it. You know that one box has twice the amount of money as the other, but you don’t know what the exact amounts of money are, nor do you know which box has twice as much money, just that one does.

Say you pick a box at random, and open it, and it turns out to have $100. You are then given the option of keeping the $100 or forfeiting it and taking whatever’s in the other box. Should you switch?

Intuitively, it makes no sense that picking a box at random and keeping it or picking a box at random and then choosing the other box would make any difference at all. But after you’ve opened the box, the expected value if you keep the $100 is $100. The expected value if you switch seems to be 0.5 x $50 + 0.5 x $200 (50% chance the other box has $50, 50% chance the other box has $200) = $125. Which means you should switch. Which is odd.

So what’s the flaw in the analysis? Or is there one?