In an earlier puzzle
, you were handed two envelopes, one of which contained twice as much money as the other. After opening one, you were given the chance to swap. At first glance, it appeared that the your chance of getting more money could only increase each time the envelopes were swapped, but clearly this was nonsense: since there is no probability distribution which allows all real numbers to have the same probability, some values would have to have been more likely than others.
Suppose instead envelopes contain the non-negative integer sums 2n and 2n+1 with probability q(1 − q)n for some fixed q < 1/2
Now of course if the envelope you open contains a 1, you know the other must contain 2, so you ought to swap.
But you can do even better than this. Suppose you open an envelope and find an amount of money 2k
What would the expected value of the second envelope be?
Does this lead to the same paradox?
Lets look at this as a bet
you open a envelope, and you find it has $2X in it
well then the other letter can contain $4X or $X
there for you always have a chance to gain twice as much as you have to loose (chance to gain $2X or lose $X).
As there is an equal chance to have either the greater or the smaller value, you would suggest that it would be better to switch envelopes, but this is not exactly the case.
For example as you opened an envelope, you are shocked to find $10 000 000. You think, i have a chance to get $20 mil and i would only lose 5.
As there are equal odds to whether you chose the greater amount, or the smaller amount, the choice is really how greedy the person is.
Mathematically, it is a good gamble. On small, such as $100, the loss of $50 would not be significant. The loss of $5 000 000, an amount you could live off, is not worth the risk, even if there was a equal chance to triple the amount and lose half.
Although, as it is money, i suppose with a significant amount, if it wasn't a cheque, you could tell by the weight 8) but i guess that defeats the purpose.
Posted by Bj H-W
on 2005-09-04 13:20:48