The original problem..."Just thought I would pose a brain teaser for discussion as a distraction from some of the more serious threads There are two unopened envelopes. The only information you have is that one of them contains twice as much money as the other. You choose one of the envelopes and you open it to find $100.You are then made an offer. You can give that envelope back and have the contents of the other envelope. The question is - Is this a good bet or not?bbbbbOK, I know it's not a poker question but it can cause lots of discussion - particularly over a few beers!PF "The answer..."(You shouldn't take into account whether or not you could do with the $100 - you should just try and decide if the odds are in your favour if you make the change.)"If you take an envelope at random then any envelope you choose can not be better than the other. So if you get $100 you basically have to decide whether you could do with the $100, that’s all there is to go by. Good job setting everyone off on the wrong track though!OK, so let’s get on with it, why the inconsistency?The answer is a bit like Einstein’s relativity, pretty simple really. We have to look at the two different points of view.After the contestant has opened an envelope and found $100…From the host’s point of view… There is no question of what is going to happen if the contestant swaps envelopes. He knows that the contestant either chose the high value or the low value, and his only job now is not to give it away I suppose, lest the player works out exactly what to do. A bit like putting out a big bet and not giving away whether you are bluffing or betting for value.From the player’s point of view… Either the two envelopes contained $150, and swapping will yield him $50, or the two envelopes contained $300 and swapping will yield him $200. What is the probability of the host giving away $300 as opposed to $150? Easy! It’s 2:1; what you would get on the roulette wheel or any gambling game. Remember that the player has no idea what the host is giving away, and the fact that he got $100 gives no information.So now…2 out of 3 times there would be $150 in the envelopes, and if he swaps he gets $50, an equity of $33.1 out of 3 times there would be $300 in the envelopes, and if he swaps he gets $200, an equity of $66.Add them up and you get $100 (if you add some cents above too).So now there is no value in the player swapping, the equity is $100 both ways.Where did we get fooled? When they said there is a 50% chance that the player got the high (or low) envelope. This is only before he opens the envelope. After the envelope is opened the different entities have different points of view. Again, from the host’s point of view, he knows 100% one way or the other. From the player’s point of view, either there was $150 in the envelopes and he got the higher (would happen 2 out of 3 times when compared with other option, there was $300 in the envelopes and he chose the lower.This means that when the host has $300 in the envelopes, there would be another 2 lives from the player’s point of view, when the host only gave away $150.Just because the game engineer himself can come out with linear choices doesn’t change things. This is a “many worlds” thing.The numbers don’t lie, this is the way it is!I would appreciate being recognised as the person that came up with the right solution first, for those people quoting it, if possible.Spikey is available for any well paying math (or physics) jobs, even though he has a good job already, he does need a change sometimes!Note that Spikey doesn’t fall for any crap that gets spattered about, doesn't write spaghetti computer programs, and generally gets it perfect every time.
If you want to die early, just retire -- Jeff Kennett