Craig Fairchild wrote:

>Let's play a game.


Okay!

>Let's call it "Find the Ace of Spades".


Sounds fun!

>We'll use a 52 card deck.

Fine...

> After the deck is randomly shuffled, you get to pick a card
> face down, and put it in front of you, remaining face down.

Okay, it's in front of me - so I have a 1 in 52 chance of
having the Ace of Spades, you have a 51 in 52 chance.

> 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.


So at this point _you_ *know* who has the Ace of Spades, just
as Mr. Hall knows which door the prize is behind.

> Now we're left with two cards, yours and mine. Since we've
> seen the other 50 cards, we know that one of our two is the
> Ace of Spades. So we both have a 50-50 chance of having the
> Ace of Spades - right? :-)


Wrong.  The odds have not changed!

With my original card I had a 1 in 52 chance and you had a 51
in 52 chance - those odds don't change just because you've
turned over 50 of the cards you have!

*Because* you know who has the Ace of Spades - that's the 'All
other things being *equal*' part - things *aren't* equal!  If
I'm allowed to choose between odds of 1 in 52 versus 51 in 52,
I'll switch!

> Being a gentleman, I'll give you quadruple your money on any
> wager you'd care to make that you have the Ace of Spades and
> not I. We can play all night if you'd like! :-)


Being a gentleman as well, I'll *take* that quadruple money 51
out of 52 tries (because I'll swap cards with you *every* time!)

Now if you wanted to add a third person to this little game, and
that person were to pick a card for each of us take to put face
down, with the guarantee that *one* of those cards was the Ace
Spades, *then* the odds would be 50-50 (because _you_ *don't*
know) - In which case, I'm not playing.   :-)

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