Musings on a whole load of stuff - see the post tags down the left side to get the flavour of shit I talk
Sunday, 13 February 2011
The 2 Envelope Problem
You are on a gameshow – a pilot episode, so you have no idea about the level of prizes likely to be offered. You answer a couple of questions correctly and are told to chose one of two envelopes. Inside the envelope you chose is $300 and the host tells you that one of the two envelopes has twice the amount of money in it than the other one. He asks you if you wish to swap for the other envelope – i.e. instead of $300 you could win $150 ($150 less than you currently have) or you could win $600 ($300 more than you currently have). It would seem to be a 50:50 likelihood of either outcome. 50% chance of losing $150 and 50% chance of gaining $300 – anyone familiar with gambling, or mathematics, will tell you that on average you will make a profit if you take this gamble therefore you should swap to the other envelope.
The problem with this reasoning is that it would apply whichever envelope you picked – the odds of your picking the envelope with the higher amount in it is 50%. swapping envelopes should still give you 50% chance of getting the higher amount. Is there any gain in swapping?
So far as I know, there is no consensus over a solution to this paradox. The "Cover's Strategy" is obviously rubbish if given a little thought - of course it works as a strategy, but it does not explain the paradox. If an explanation does not exist then which predicate is false? This has troubled me for over 10 years.