3 ms·
The mistake is in step 7. You cannot calculate the expected value (of the envelope you get, when you pick one randomly and then swap) like that, as A is not a
by Gehinnn 4y ago
The mistake is in step 7.
You cannot calculate the expected value (of the envelope you get, when you pick one randomly and then swap) like that, as A is not a constant and actually depends on if you took the less or more valuable envelope initially.
If you use A1=X in case you took the less valuable envelope initially and A2=2X in the other case, the expected value is
(1/2)*(2*A1) + (1/2)*(A2/2) = 1.5X
and if you don't swap, you get:
(1/2)*A1 + (1/2)*A2 = 1.5X
... which makes sense.
If you change the experiment so that the value of the second envelope does not depend on the first one (if you first chose and only then the value of the other envelope is randomly either double or half), A is constant and you actually should swap.