A rather related thingy I thought up:
Let us take the two-envelopes situation and make it into a game, where you can
keep switching over and over, with a new set of envelopes every time (which are based on your
current envelope).
You start with $128. At the beginning, the two unknown envelopes, therefore, contain
$64 and $256. As already stated, we expect an average gain of 1/4
our current value every time.
So, keep switching a large number of times, and you'll be in the money!
We could also model the game as follows: You have a "ladder" of money
values, with each rung two times the one below it. Each time, you are asked,
"Hit or stay?".
Stay means that you are done playing the game, and you wish to take your money.
Hit
means that you have a 50% chance of going up on the ladder, and a 50% chance of going
down on the ladder.
However, after a large number of hits, don't we expect an even
distribution of ups and downs? If that's the case, we won't be really
going anywhere on the ladder. So, we don't have any expectation of gain or loss.
Quite contradictory.
Even more interesting: What if we had a 60% chance of losing half (moving down on the ladder), and
a 40% chance of doubling (moving up on the ladder)? According to our first model, we should have an
average exponential gain of 10% every time, but the second model says that
our money would exponentially decrease on average!
So, which model is correct? :-)