Two envelopes problem
> Having chosen an envelope at will, but before inspecting it, you are given the chance to switch envelopes. Should you switch?<p>Normally we would assume there is someone who owns the envelopes of money. Now their incentive could be to offer to switch only if the envelope you chose contains more money (so they have a chance of keeping more). More like Monty Hall.<p>Even without that, after switching, should you switch back? The equation given argues you should switch back, so obviously the equation must be wrong.<p>It is easier to think: if I switch I have a 50% chance of making $X and a 50% chance of losing $X. Switching doesn’t change expected $.<p>I actually think the later problem in the Wikipedia article is more clever:<p><pre><code>  Two people, equally rich, meet to compare the contents of their wallets. Each is ignorant of the contents of the two wallets. The game is as follows: whoever has the least money receives the contents of the wallet of the other (in the case where the amounts are equal, nothing happens). One of the two men can reason: "I have the amount A in my wallet. That's the maximum that I could lose. If I win (probability 0.5), the amount that I'll have in my possession at the end of the game will be more than 2A. Therefore the game is favourable to me." The other man can reason in exactly the same way. In fact, by symmetry, the game is fair. Where is the mistake in the reasoning of each man? — "Martin Gardner: Aha! Gotcha"</code></pre>