Two envelopes problem
It is very simple.<p>Let&#x27;s represent the poorer envelope by the value 0.5 and the more bountiful one by 1.0.<p>The expected value from the envelope draw is 0.75: 0.5 drawn with a 50% probability contributing 0.25 to the expected value, and 1.0 drawn with a 50% probability contributing another 0.5 to the expected value for a total of 0.75.<p>If you were to do a large number N of these draws, the value you will obtain will be from a distribution that is centered on 0.75 N. So far so good?<p>So the way to look at a draw is this: you&#x27;re going to be getting the expected&#x2F;average value of 0.75, together with either a 0.25 bonus if you pick the better envelope, or a else  -0.25 penalty if you pick the poorer envelope.<p>So you see, those two situations are equal and opposite. By switching, you either turn a 0.25 bonus into a -0.25 penalty (- 0.5) or vice versa (+ 0.5) with equal probability.<p>If you need any more complex analysis than this, you need to be thwacked on the head.<p>The rhetoric about doubling versus halving the money is a distracting red herring. If you switch from the 1.0 envelope to the 0.5 envelope, you&#x27;re down 0.5, and if you switch the other way, you&#x27;re up 0.5. That&#x27;s it.<p>Accounting is based on debits credits not halving and doubling. You would never update a ledger by crediting double some amount to one account, and debiting half the amount from another account. It&#x27;s all purely additive.<p>You have to look at what you&#x27;re potentially gaining or losing <i>on its own</i> not as a fraction of some guaranteed fixed portion that you&#x27;re getting from either envelope.