Craig writes:

> Let's play a game.
>  Let's call it "Find the Ace of Spades".
>  We'll use a 52 card deck.
>  After the deck is randomly shuffled, you get to pick a card face down, and
>  put it in front of you, remaining face down.
>  I'll take the remaining 51 cards, examine them, discard 50 of them, face up
>  so we can both see them, and then place the remaining card face down.

It is Craig's small phrase "examine them" that completely changes this game
from the one that Denys presented. If the 50 cards in Craig's game that were
discarded were sequentially and randomly drawn -- and each one in turn proved
to be NOT the Ace of Spades, then the "information" (probability of success)
associated with the remaining cards goes up proportionally on every unknown
card, at exactly the same time, at exactly the same rate.

At that point in time where there are 20 unknown cards left, your chance of
holding the ace is 1 in 20 -- but that's exactly the same for every other
card also. Clearly, the probability that you don't hold the winning card is
19 in 20, which are obviously not good odds. But given fair draws -- and
still a lack of success up to now in finding the ace, your probabilities of
holding the ace increase, but they do so at no greater or lesser rate than do
any other specific card's.

However, given the presence of a "deus ex machina" (in this case, it's Craig
that takes on god-like qualities and rearranges the odds to foil the lives of
mortal men), the chance that you have the ace is 1 in 52 -- and that didn't
change with Craig selectively showing you the other 50 cards. He originally
held 51 out of the 52 cards, and he now selectively shows you 50 cards which
he *knows* not to be the ace.

Probability and information theory are intimately linked ideas (as is
thermodynamics; one of the oddest and hardest ideas that Claude Shannon had
to demonstrate to people was that you have to "pay" for information with
actual, physical heat units).

In this case, the question of overriding importance is: "What did Craig know
and when did he know it?" If he has to pick his card at the same time you do
-- and he finds out which cards are not the Ace of Spades at the same time
you do -- he's no better off than you are. If, on the other hand, he gets the
whole deck, minus the one you hold -- and then is allowed to "pre-examine"
them -- you're basically sunk 51 out of 52 times if you hold onto your
original card.

Wirt Atmar