-----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@pobox.com