> -----Original Message-----
> From: Joshua Singer
> Sent: Thursday, December 04, 2003 4:58 PM
> To: math-fun
> Subject: Re: [math-fun] re: The Cooler & math fun at the casino
>
> This is the only point I was making in my initial response.
> When S draws a card,
> it certainly doesn't affect
> A's expected return on the round in question.
True. But it can affect the expected return in the following
round.
>
> The argument that S by his play makes a hot deck end sooner
> seems valid, but I'd
> need to see a careful argument,
All you really need to prove that the other player can hurt
A's chances is an example situation where this is the case.
Let's suppose that there are 2 cards left in the deck, both red.
Without the other player, A gets 2 guaranteed wins against the
house. If the shill plays the next hand, A gets only one guaranteed
win. So it's pretty clearly in the house's favor to have the
shill play this hand.
While this is theoretically interesting, I don't believe this
is actually used as a countermeasure by casinos in the
real world. In a hand-held game, you can achieve the same
effect by having the dealer count, and shuffle early if
the deck becomes favorable to the player. This technique,
known as "preferential shuffling" is used by casinos. In
a shoe game, where the time to reshuffle is fixed in advance
(by inserting a plastic "cut card" into the shoe, and shuffling when
this point in the shoe is reached), the house could profit by
having a shill who played when the deck was favorable. But this
uses up a seat at the table, and is an extra salary to pay.
Doing this at every high-stakes table would be overkill. And
if you're going to find the advantage players, and use this
countermeasure only at their tables, you have to detect
the skilled players. And once you've done that, why not just bar
them?
>
> Here's an interesting related problem: Consider the
> following version of
> red/black: you post one dollar up front,
> and then the dealer will start rolling the deck one card at a
> time until you tell
> him to stop. At that point, the house pays 1-to-1 if the
> *next* card is red;
> otherwise, you lose. Surprisingly, this is a fair game.
> That is, you cannot make money at it.
The easy way to see this is true: When you say to stop,
the order of the remaining cards in the deck is random.
So it wold be the same expectation if, when you placed the
bet, the dealer dealt the next card from the bottom
instead of the top. Now the game is "you win if the bottom
card is red, and lose if it's black, but before you
make the bet, you can look at cards from the top of the deck,
as many as you like". As long as you are ultimately required to
bet on the bottom card every round, it makes no difference
whether or how many cards you look at from the top of the deck.
Andy Latto
andy.latto(a)pobox.com
>
