The two-envelope paradox
Asked at Jane Street
Two sealed envelopes are placed in front of you; one contains exactly twice as much money as the other. You pick one, open it, and find dollars. You reason:
The other envelope holds either or , each equally likely. Its expected value is , so I should switch.
But by symmetry the same argument applies before you look at all, and after switching you'd want to switch back forever.
Where exactly is the reasoning wrong?
Show a hint
Write down what the symbol actually denotes in each of the two branches "the other is " and "the other is ." Is it the same number in both?
Your answer
This one is open-ended. Work it through, then check your reasoning against the full solution.