Should you switch if you never look?

Question

Two envelopes hold money, one exactly twice the other. You pick one but do not open it. A friend offers to let you swap for the other envelope, still sealed.

Does switching change your expected winnings, and if so by how much?

Accepted Answer

No, it changes nothing, and the cleanest way to see this is symmetry. Let the two actual amounts be x and 2x. Before you look, your envelope is equally likely to be the smaller or the larger, so its expected value is 	frac12(x) + 	frac12(2x) = 	frac{3}{2}x. By the very same reasoning, the other envelope also has expected value 	frac32 x. Swapping one 	frac32 x-envelope for another 	frac32 x-envelope leaves your expected winnings untouched, so the expected gain from switching is 0. The "expected value is 	frac54 of what I hold, so switch" argument fails because it fixes your own amount …

Should you switch if you never look?

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