Permalink Submitted by Edward on November 24, 2017

x is the amount in your chosen envelope. Let L = lower amount and H = higher amount. H=2L.
Expected value of x is 1/2*L + 1/2*H = 1/2*L + 1/2*2L = 3/2 * L
Expected value of the other envelope is also 3/2 * L by symmetry, so there's no point in switching.

The (deliberate) flaw in the formula they set out is that they say the amount in the other envelope is either 2x or x/2, but this means that one envelope is worth 4 times the other, which we know isn't the case!

## Two envelopes

x is the amount in your chosen envelope. Let L = lower amount and H = higher amount. H=2L.

Expected value of x is 1/2*L + 1/2*H = 1/2*L + 1/2*2L = 3/2 * L

Expected value of the other envelope is also 3/2 * L by symmetry, so there's no point in switching.

The (deliberate) flaw in the formula they set out is that they say the amount in the other envelope is either 2x or x/2, but this means that one envelope is worth 4 times the other, which we know isn't the case!