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Rhythmic Rolling and the Gambler's Jamboree

13 December 2003

This is the last of my four-part series on craps and deals with some mathematical issues related to the subject of rhythmic rolling. This is a term that refers to controlled shooting, specifically, setting and delivering the dice in a manner that increases the frequency of a seven occurring on the comeout roll and reduces the frequency of a seven occurring the rest of the time. Many people are skeptics and doubt that this can be done and others flatly state that it is a bunch of hooey. In particular I have been asked how, as a mathematician, I can buy into such a notion. Let me address this.

Originally, I was one of the skeptics. Then back in September of 2002, while attending the G2E convention in Las Vegas, I was having dinner at the Hilton with Frank Scoblete and John Grochowski when Frank suggested that we shoot some craps after dessert. John had other obligations so Frank and I headed for the craps tables. While we were playing Frank suddenly said "Sharpshooter is at the next table. Let's go." I had heard of this guy, Sharpshooter, but had never met him. The whole story is related in my review of Sharpshooter's book which appeared earlier on this site. Suffice it to say that the guy was impressive. Still, I left unconvinced that an average Joe like me could develop a shot controlled enough to make any money playing craps.

I had done some mathematical work for Frank and as payment Frank gave me a free entry into one of his Golden Touch craps seminars; this one was held in Norwich, Connecticut this past August. I attended though I admit it was mostly out of curiosity. What a surprise was in store for me. The instructors, Bill Burton, Dominator, Bob (Mr. Finesse) Convertito, and Randy (Tenor) Rowsey were top notch and clearly my shooting improved over that two-day course. The thing that put me over the top, so to speak, was the no-sevens contest that we had at the end of the course. Each student had two chances to throw as many rolls as he or she (AP, Frank's wife, was one of the students) could without rolling the dreaded seven. One student, whose name was Gary, clearly had the best delivery. The dice consistently came rolling out of his hand in unison, staying together, and landed softly a few inches from the end of the table. It turned out that Gary had taken the Golden Touch course a year earlier and had returned to take it again and sharpen his skills. He told me he had practiced every day for a year. Believe me it showed. He rolled 31 times before the seven showed and was the best in our class; yours truly only made 12 rolls before the 7 but that is better than average (6 rolls). I had seen, with my own eyes, a student become a really skilled shooter. I was a believer. I bought a practice rig and have thrown every day that I could since then.

Two weekends ago, November 14th, 15th and 16th, Frank Scoblete organized a Gambler's Jamboree at the Gold Strike Casino Resort Hotel in Tunica, Mississippi. Frank had called on several gambling experts to give talks and hold clinics for the attendees. Among the luminaries was Bill Burton (Low Limit Hold'Em Poker), Henry Tamburin (Blackjack Expert), John Robison (Slot and Video Poker Expert), John Grochowski (Chicago Sun Times writer and Video Poker Expert), Frank Legato (Gaming Columnist and Slot Reviewer - a really funny guy), Jean Scott (Video Poker and Comps Expert as well as the keynote speaker), Walter Thomason (Blackjack Writer), and the Golden Touch craps crew. Oh yeah, I was there too. Over 300 people attended and I would call it a huge success.

Two things happened in Tunica that reinforced my belief in controlled shooting. First, on Saturday night after dinner, Frank and several of the Golden Touch crew headed next door to the Horseshoe to shoot some craps. Several of us tagged along including Jean Scott, Linda Mabry (more on Linda later), and myself. To make a long story short, when Frank got the dice he held them for an hour! Everybody at the table made money. People were standing three deep to try to get a spot at the table; there were none to be had, we all stayed. I estimate that table paid out between $10,000 and $20,000 during that roll. It was incredible. My biggest mistake was not betting enough (Isn't hindsight great?) but I still made a lot of money. What a memorable event.

Second, on the next day as the conference was winding down and the Golden Touch crew had some spare time, I asked Mr. Finesse if he would take a look at my shooting technique. Although all of the GTC gang are great instructors, Bob was my personal instructor in Norwich and I have a special affection for him and his astute observation of shooting. He said fine and did have one criticism; my backswing was too fast. Other than that, however, he said that my shooting had improved and he basically just watched me. Unbeknownst to me he was keeping count of my shots and it turned out that I rolled 26 times before hitting a seven. I was not eligible to be in the no-sevens contest that was being held because I was a speaker, but had I been eligible I would have come in second (I believe the winner had 28 rolls). So I guess I'm learning. I have no illusions though; I know I need a lot more practice before I can call myself a controlled shooter.

What is the mathematical story when it comes to controlled shooting? One way to quantify the skill of a shooter is to calculate that shooter's sevens to rolls ratio or SRR. This is just the average of the number of rolls the shooter has between sevens. For a completely random roller this number would be 6 since, on average, a shooter will roll a seven once every six times. A skilled shooter is one who can raise that number. Let's see what various SRRs do for the shooter. To simplify matters I am going to just look at placing or buying the 6 and 8. First we'll discuss placing.

To place a number you hand the dealer your chips and tell him you want to place that number. The bet is that the selected number will occur before the seven. The wager is similar to the odds bet in craps, but it is not paid at fair odds. Fair odds on the 6 or 8 are 6 to 5; placing either of these pays 7 to 6. Clearly the expected return on this wager is

(1) Exp = (5/11) x (7/6) - (6/11) x 1 = - 1/66

which represents a house edge of 1.515151…%. This is a reasonable bet but not a great bet. Let's see what a controlled shooter can do with this wager.

To make matters a bit easier, let me just use the symbol r for the SRR. It then follows that the probability of a seven is just 1/r. In general this will be a smaller number than the random case in which r is 6. I will assume that the probabilities of the rest of the numbers increase in a manner that keeps their relative frequencies the same. That is, a three will occur twice as often as a two, a four will occur three times as often as a two, and so on. This assumption is probably not correct. Depending on how the shooter sets the dice and his individual nuances in shooting, some numbers will probably occur more frequently than others. The assumption probably best approximates the average situation over the population of shooters with a particular SRR.

Very well, if the probability of a two is p, then the probability of a three is 2p, the probability of a four is 3p, and so on. If we add up the probabilities of all of the numbers except seven we obtain 30p. Since the probability of a seven is 1/r, we obtain the following relation

(2) 30p + 1/r = 1

This equation can be solved for p as follows

(3) p = (r - 1)/30r

It follows from this that the probability of rolling a six or eight is just (r - 1)/6r. Using the fact that the probability of a seven is 1/r, along with the observation that the probability of making a point of six is just the probability of rolling a six divided by the sum of the probabilities of rolling either six or seven, we have the following formulas

(4) Prob. of Making Six = (r - 1)/(r + 5)

(5) Prob. of Seven Out = 6/(r + 5)

For a player placing (say) the 6, the expected return is simply

(6) Exp. = 7(r - 1)/(r + 5) - 6(6/(r + 5))

which, when divided by 6 (the amount risked), reduces to

(7) Exp. = (7r - 43)/(6r + 30)

Clearly the player has an advantage whenever 7r > 43 or r > 43/7. The number 43/7 is approximately 6.1428. In other words, if a shooter has an SRR greater than 6.1428 he has an advantage when placing the 6 or 8. A player with an SRR of 7 has, according to (7), an 8.333…% advantage over the house. Not bad!

I traveled to Tunica by car. My wife and I had spent two months in Florida and decided to simply return home to Amherst, Massachusetts by way of Tunica. This meant our trip home was 2400 miles rather than 1200. It was worth the extra miles though. For, not only did we enjoy the Jamboree, but we got a chance to have lunch at Wintzell's Oyster House in Mobile (best raw oysters I've ever had) and spend a night in Biloxi and have diner with the very charming Linda Mabry. Michael Shackleford (The Wizard of Odds) had been lecturing in New Orleans and drove over to Biloxi and joined us for dinner as well.

Linda is the gaming writer for the Biloxi/Gulfport Sun Herald and is very knowledgeable about gaming in that area. One of the really interesting things she told me is that all of the casinos in that area allow you to buy the six or eight for $1 on a $30 wager and the vig is only charged on winning bets. This means that your $30 wager wins $35 rather than $36 on a winning roll. A simple calculation shows that the house edge for this bet is 1.5151515…%, exactly the same as a place bet. But wait, there's more. All of the casinos except Beau Rivage, Treasure Bay, and the President allow you to buy the six or eight for $1 on a $35 wager and the vig is only collected on winning bets. This means that $35 returns $41 rather than $42 on wining bets. This brings the house edge for a random shooter down to 1.2987%. Let's see what this does for a controlled shooter.

Doing calculations similar to those in (5) and (6) above we obtain the following formula for the expected return per dollar risked when buying the 6 or 8 for $1 on a $35 wager:

(8) Exp. = (41r - 251)/(35r + 175)

Clearly the player has an advantage whenever 41r > 251 or equivalently r > 251/41. The number 251/41 is approximately 6.122 so for this wager a controlled shooter has an edge over the casino whenever his SRR is larger than 6.122. For a player with an SRR of 7, the player, according to formula (8), has an edge over the casino of approximately 8.57%.

Controlled shooting is a reality; I've seen it with my own eyes. If you are willing to put in the time and effort you too can be a controlled shooter. Don't expect instant success though. See you next month with a lottery scam that you'll find hard to believe.

Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers