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Probability: Now for the Basic Arithmetic of Card Counting…

The advantage player who dresses like a bum (or worse) has it all worked out, in part with the help of a computer at home

In “Can a good hustler count cards like a computer?” (podcast), Walter Bradley Center director Robert J. Marks continues his discussion of card counting techniques with gambling ace Salvador Cordova, also a mathematician and engineer. An “advantage player,” Cordova made his living, in part, by beating the casinos from about 2005 through 2014. Note: This podcast involves a fair amount of discussion of specific numbers so the partial transcript below may be especially useful:

This portion begins at approximately 12:00 min. A partial transcript and notes, Show Notes, and Additional Resources follow.

Robert J. Marks: That’s the card counting system you use, Omega II?

Sal Cordova: Right. If you see an ace or an eight, you just add zero to your running total…

The dealer is dealing and you observe aces and eights from his deck. I’ll just add zero, which is easy. If you see four, five or six, you add the number two…

You have to try to separate the numbers. And then, in all of that, the dealer is calling out numbers like you have 16 or… So that’s the first skill: Not to be confused.

So aces and eights are zero. Two, three, and seven, you add one. Four, five, six, you add two. Face cards, which are like tens, jack, queen, king, you subtract two. And nines, you subtract one. And that’s it. That’s one of the best card counting systems in terms of accuracy.

And then you can do what they call… If you want to keep two counts in your head, you count the number of aces that have come out. When you can keep those tallies, you have some idea of then the remaining strength of the deck.

Here’s a vid again…

Robert J. Marks: So what do you do with that number?

Sal Cordova: If it’s a very high number, you start increasing your bet. if it’s a low negative number, you want to reduce it…

But then you have to adjust it by the number of cards dealt out. I think you call it the true count. So let’s say you have a single deck game, that’s the easiest. He’s dealt out half and let’s say your running count is at six. You divide six by a half. Now it’s at twelve. That’s when you bet your maximum.

With each hand, I mean with each count level, there’s an associated advantage. I think the maximum advantage you might have when the counts are that high could be like 6%. So then that’s when you had the computers actually figure out beforehand how much you’re going to bet.

The computers? What computers?

Sal Cordova: Before I would go into the casinos, I’d look at my bankroll, like say it’s $30,000 and the computer would say, “Okay, if you’re operating at a Kelly fraction of one eighth, this is how much you can entitle yourself to when you have the maximum count.” And you’d go into the casinos and say, “Okay, if I have a count, a true count of six, this is how much… I could push out $300. If a true count of three, I would push out this amount.” You have all these tables you would memorize and that’s how you would use it.

Sal Cordova

So the first skill is to actually be able to count the decks and keep that running count. And then you have to divide it by the number of cards dealt out. And you can either use your eyes to kind of just guesstimate or you could actually count also the number of cards that were dealt out. If you really wanted to be an aficionado, you could also be counting the aces and you could kind of adjust the counts with the number of aces dealt out…

Robert J. Marks: You’d have to be sharp when you did this, you couldn’t be sleepy or tired.

Sal Cordova: It would be undesirable. And the thing you did not do is drink any alcohol. Of course I don’t drink. I rarely drink anyway, but that used to be kind of the running joke. If you see someone drinking bottled water…

Robert J. Marks: I wonder if the casino bosses look at that.

Sal Cordova: Oh yeah. It’s like, “This guy looks like a scholar and he drinks bottled water. And he doesn’t seem to be bothered whether he loses or not…

Robert J. Marks: And by the way, card counting doesn’t guarantee you win; it just increases the probability you win.

Sal Cordova: Right. So you don’t over bet because you try to get the Law of Large Numbers, in your favor. And the ones that are good at it love the game just for the challenge. Bill Gross, that hedge fund manager of a trillion dollars, likes to play blackjack. He’s not doing it for the money.

He has to sneak into the casinos now because they know who he is. And sometimes you can get away with it. You have to go in there in a disguise. When I started to get photographed, my favorite outfit was the pimp outfit… Loose clothing and I’d have to walk like a boy from the hood. Now the getup that they said looked at least halfway convincing was, I’d be in my cargo pants and wearing a deer hunter outfit. That kind of made me blend in or it just kind of looked like you’re just a guy who’d been… Some of these workmen who paint houses or whatever.

Robert J. Marks: A good old boy.

Sal Cordova: Yeah, a good old boy. And I had some partners there that tell me, “Yeah, that looked pretty convincing until they heard you talk.” …

Can a computer do it all?

Robert J. Marks: Let me ask you a question. Clearly there’s different levels of card counting depending on how complex things you can juggle in your mind. And as you go up the difficulty, the chances of you winning are better, right? So that suggests that there is a best way of doing it. Now that would require you to literally be a computer… Could you really clean up at card counting if you had an interface to a computer program which told you the optimal play, as you begin to type in all of the cards that have been played?

Sal Cordova: No. No.

Robert J. Marks: That’s a surprising answer.

Sal Cordova: No, because [of] the merging of human and computer intelligence. The computer intelligence gave you the human strategy to play it. But as I said, the correct plays are still 75%. The correct plays being, you memorize all of these tables. And I had pages of tables memorized where it would tell you, “Okay, under this count, this is the best play to make.” And it would be 75% of what the computer could do. And in that book, I loaned you the book but…

Robert J. Marks: Okay. Let’s let me talk about this. Sal gave me a book called The Theory of Blackjack and it’s in its sixth edition. And the subtitle is, “The complete card counter’s guide to casino game of 21 by Peter A. Griffin. Did you learn from this book?

Sal Cordova: No. I learned from Blackjack for Blood by Bryce Carlson. But that book was theoretical. It elaborated on Edward Thorp’s original work. Very complicated counting system. And it tested out well, but it was just, it was brutally difficult to use in the casino.

Robert J. Marks: So what you’re saying is that this simple card counting algorithm that you came up with is pretty close to as good as you can do?

Sal Cordova: Right. Now you did mention about these computers. There was some long time ago, Keith Taft, who built the first wearable computer and he would use his toes to type the cards that he was observing. And then it would buzz when the counts were high and tell him… And he would play. He was successful at that.

And then he teamed up with some … not the most savory characters. He would have these people as observers with the computers and then the big whales, the big bettors, team up. And the big betters would rely on the guys with the computers just standing behind them. The guys with the computers would let them know when to start raising their bets.

And they would play probably basic strategy or some variation of the strategy. And they were cleaning out for a month until the casino surveillance figured it out.

“Taft was a real-life Inspector Gadget. He was a legitimate electronics genius who devoted
roughly 30 years to developing devices that defeated the casino.” (4:22 through 8:20 min)

Why it’s hard to make a living gambling even if your math beats the casino’s

Robert J. Marks: Usually the casino, if you figure out something’s going on, they change the rules so that you can’t game it, if you will.

Sal Cordova: Right. And that resulted in a Las Vegas law that you can’t bring computers into the casino.

Robert J. Marks: You can’t, but what’s interesting is you can bring cell phones in and you know that those cell phones are more computationally powerful than anything Claude Shannon could have brought in there.

Sal Cordova: So I think what they would say is you can’t use it in the commission of the game, you could get in big trouble because you could still beat the game if you’re 75% efficient. Most of my professional gambler buddies would say, “Don’t even try. The risk of you getting thrown in jail and prosecuted, even if you’re innocent, just don’t even give them an opportunity to hassle you or prosecute you.”

Robert J. Marks: There goes my get rich quick scheme. Well, Sal, this has been fascinating. We’ve been talking to Sal Cordova. The guy has more degrees than a circle. He has degrees in mathematics and computer science, electrical engineering, a master’s in physics, and he’s done a lot of graduate work in biology. Could you have made a living [gambling]?

Sal Cordova: If they let me keep playing, yeah. I would have retired, like, in a year, because of the Exponential growth law. You could be doubling your bankroll every few weeks.

Exponential growth law demonstrated with ping pong balls

Robert J. Marks: Yeah. There’s the old story about the king that rewarded somebody for some reason by putting rice on a chessboard where he put one grain of rice, then two, then four than eight, and that’s the payment that the guy asked for. And then they found out when they got to the 64th square, that that was more rice than existed in the world. That exponential increase is just astonishing. Well, we’re going to talk next time about [card counting] Christians.

Note: There’s a movie about the card-counting Christians, Holy Rollers (2011):

Stay tuned for Marks’ and Cordova’s discussion of Holy Rollers and gambling.

Note: This information is intended for math nerds and people entertained by probability theory and statistics. It is not directed at anyone who has a gambling addiction. For help, see Gamblers Anonymous.

Note 2: The featured photo is courtesy  Amanda Jones on Unsplash.

Next: Card Counting Strategies and Dangers (podcast). Breakout stories to follow.

Here are the writeups (with partial transcripts, notes, and links) of all the podcasts with gambling maven Sal Cordova:

Gambling: WHY the house always wins in the long run… The casinos are not cheating. They rely on the Law of Large Numbers, part of the mathematical structure underlying our universe. Robert J. Marks and fellow engineer and mathematician Sal Cordova look at the many creative methods gamblers use to improve their odds.

Casinos: How nerds gamble and win, using the Law of Large Numbers The American Physical Society created Las Vegas’s worst week in history and Don Johnson cleaned out Atlantic City. How? Sal Cordova explains to Robert J. Marks how nerd gamble and win, mainly by deciding whether to play at all and, if so, how to manipulate the house’s strategies.

Croupier hands dealing cards on t blackjack poker table, gambling table with cards and chips

The struggle between casinos and advantage players The scene is enlivened by assorted other characters who use romance to help in the struggle for a big win. Sal Cordova explains. Often the advantage player, who astounds us by seemingly beating probability, is taking advantage of the casino’s upper management’s weaknesses.

Can casinos ban would-be customers who might get TOO “lucky”? Sal Cordova was good enough at card counting that his photo was circulated and the casino nabbed his driver’s licence… But as casinos slowly eliminate loopholes, “math whiz” advantage players seem to find others.

Can you really be a card counter without resorting to magic? Math nerd (and successful gambler) Salvador Cordova explains how card counters improve their odds in blackjack. Successful card counting strategy originated in a theory out of Bell Labs and also found its way into hedge fund management.

Probability: Now for the basic arithmetic of card counting… The advantage player who dresses like a bum (or worse) has it all worked out, in part with the help of a computer at home. Sal Cordova tells Robert J. Marks that mathy pro gamblers could retire after a year, using the Exponential Growth Law, but casinos spot them and kick them out.

Gambling: And when advantage players team up, dealer beware! On the other hand, the movie industry has made a good thing from films of the legendary exploits. Sal Cordova details for Robert J. Marks odd strategies he and other advantage players use to win via the Law of Large Numbers — until they’re found and ejected.


The Holy Rollers: Christians who gamble for God. Not only have many successful players been Christians, probability theory was developed in part by a philosopher who became a devout Christian. Strange as it might seem, the successful Rollers were not really “gambling”; they followed the Law of Large Numbers — and the computer — obediently.

Show Notes

  • 00:06 | Introducing Sal Cordova
  • 00:36 | What Does It Take to be a Good Hustler?
  • 01:45 | A Lesson In Card Counting
  • 15:07 | You Have to Be a Good Actor
  • 16:58 | How To Use Your Teeth As a Keyboard
  • 19:28 | Blackjack Books
  • 21:56 | If You Try to Beat The Game, You Could be Arrested

Additional Resources

Podcast Transcript Download

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Probability: Now for the Basic Arithmetic of Card Counting…