[extropy-chat] Monty Hall Paradox ... Not!

[Rob Wilkes]

The Monty Hall Paradox is not a paradox.  Think of it from Monty's POV.  MH 
must have an envelope to swap you for each time that he offers a trade.  In 
the worst case (for Monty) he would keep offering you an envelope 
containing twice as much and you would keep trading.  For 5 trades the 
progression would be: $10 (start), 20, 40, 80, 160, 320. You  started with 
$10 and walked away with $320.  In the best case for MH he would keep 
offering you an envelope containing half as much and you would keep 
trading.  For 5 trades the progression would be: $10 (start), 5, 2.50, 
1.25, 0.625, 0.3125.  You started with $10 and walked away with $0.3125.

Your luck could go either way 50:50 on each trade.  It is essentially a 
random walk along a 1-dimensional integer line where the value at each 
integer is $10 x 2^i. (i being the integer, and 0 being the starting 
point.)  The number of trades determines the liability MH is at risk for: 
$10 x 2^n, where n is the number of trades he will offer you.  If Monty 
will offer you 8 trades, starting with $10, then he must be prepared to 
fork over $10 x 2^8 = $2560.00 in the worst case.  The probablilty of you 
getting that lucky is 1/(2^8) = 1/16. For 8 trades you have an equal chance 
of walking away with any of the following: $2560, 1280, 640, 320, 160, 80, 
40, 20, $10, 5, 2.50, 1.25, 0.625, 0.3125, 0.15625, 0.078125, $0.039063.

You can only get rich by being lucky and trading a lot.  You can only get 
very poor by being unlucky and trading a lot.  If you don't trade at all 
you have $10 to buy some beer for you and your friends after the show, and 
that is the only outcome that is guaranteed.

Rob Wilkes