Dice Setter Precision Shooter's Newsletter
Welcome to a special end of summer, double edition of the Precision Shooter Newsletter!
I have to warn you that one of the articles in this issue of the newsletter has some fairly serious math in it. Don't let that scare you! Even if the math itself doesn't interest you, there's some interesting theory in the article. You see, it's important to remain open minded to new ideas coming from new voices in the dice influencing community. New and creative thinking pushes the boundaries of what we know about our art form.
In other news, I've continued to make minor changes to the message board as many of you know. If you haven't stopped by lately, make a note to do so. There have been some excellent discussions recently on Discipline and Come Betting and several other great topics.
Zen Craps - Going With the Flow
For those of you who dont know Buddha had a name. He was born Siddhartha Gautama in the foothills of the Himalayas at around 560 B.C. The son of a king, Siddhartha was raised in a pleasure-palace that shielded the him from human misery and suffering. But one day the young prince saw four things that filled him with sorrow. He saw a sick man, a poor man, a beggar, and a corpse. Siddhartha was so stricken by these images that he abandoned his pampered lifestyle and dedicated himself to a life of extreme personal denial. He fasted until he was so thin that he could stretch his hands around his waist, yet he could not find the inner peace he sought. Then one day he overheard a music teacher telling his students, If the instruments strings are too tight then it will not play harmoniously. If the strings are too loose it will not produce music at all. Only the middle way, not too tight and not too loose, will produce music.
This overheard conversation changed led to his awakening, or Zen. And that in turn led him to The Middle Way and the Eightfold Noble Path. Its eight tenets are right understanding, right thought, right speech, right action, right livelihood, right effort, right mindfulness, and right concentration. Those eight tenets plus a little fortune cookie humor - were the launch pad for the Twelve-fold Tao of Craps article.
In a sense, many precision shooters are practicing a sort
of Zen. We often talk about right understanding (knowledge of the game), right
thought (approaching the game with a positive mental attitude), right speech (the language
of the game), right action (take me down), right livelihood (dont quit the day job),
right effort (give it a fair shot before saying it wont work), right mindfulness
(play when youre fresh not when youre tired or distracted) and right
concentration (shooting from the zone). In truth, though, the casino is
not a holy place. Tobacco smoke is far removed from the sweet incense of the East,
and those skimpy-costumed cocktail servers are anything but saffron-robed monks.
Upcoming SeminarsThere are several seminars coming up in the next few months. Click the links for more information.
Heavy Returns To Tunica! - Sept. 19 to 21, 2003
Heavy and Dicecoach's Right or Wrong Weekend in Las Vegas - Nov. 7 to 9, 2003
Dice Feminique! Hosted by the DiceCoach, Michael Vernon & Debbie"Soft Touch" G. - Nov. 14 - 16, 2003
Precision shooters usually discuss controlling the outcome of a pair of dice. That is, they can target the outcome of the dice taken as a pair. Lets take a look at what happens when you can control one die in a pair. Suppose you were able to throw two dice and make one of the faces show a 5. How would you bet in order to take full advantage of your skill? Which number would you choose to throw? And, assuming you couldnt hit a 5 every time, what percentage of rolls would you need showing a 5 in order to break even with the house? How long would it take to prove that your skill was significantly different than a random roller? I answer all these questions in this article. You might need to get out the old statistics textbook to follow the entire explanation. The conclusions, however, are noteworthy.
Determine which bet to
After practice, a precision shooter has developed a skill to throw a single number more than another. Of the six numbers to choose from, which number(s) will give him the greatest advantage over the house?
First we will find for each target number the optimal bet associated with it. In the comparison of the six available numbers to choose as the target number, we can treat the 3 and the 4 equivalently since house payoffs on the outcomes are identical. We can do the same for the 2 and the 5 as long as we conclude that the bet on the 2-1 combination is not optimal. With a little gambling experience, you will find out that it is never wise to bet on the 2-1 combination. That is, unless, you are trained to throw a 2 and a 1 at the same time. The 2-1 is just a bad table bet.
That said, for the target number, list all the possible outcomes of the pair of dice. Here are the possible combinations of the dice if you choose 3 as your target:
Then, given the possible outcomes, find which table bet maximizes player edge. In this scenario, you would choose to play the Hard 6 over any other bet. There is one 3-3, no Easy 6's and only one 7. The probability of hitting the Hard 6 is thus 50%. Casino odds are 9-1, resulting in a player edge of 400%.
Similar analysis is done for the 1,2 and 6. Below is a list of the target numbers, their optimal table bet and the player edge.
Breaking Even with the House
Obviously no one is skilled enough to guarantee his or her target number on every throw. If we were, we would enjoy the wonderful player edge listed above. What if the shooter could only guarantee the target number once in every 10 throws? Would he still have an edge over the house? How about once every 50 throws? Every 100? How many times does a shooter need to guarantee his target number in order to break-even with the house?
Define j such that 1 out of j times, the shooter can guarantee his target number. Our goal is to find the number of throws, j, at which point the house has 0 edge over the player.
First we will need to find the probability of rolling an outcome (a Hard 6 for example) given that j-1 times he is a random roller, and then the jth time he guarantees the target number. So let m = probability of outcome given a random roll (for the Hard 6 it is 1/11), and let n = probability of outcome given target number rolled (for the Hard 6 it is 1/2).
Then the weighted probability P = ((j-1)/j) * m + (1/j) * n.
A precision shooter guarantees that 1 out of 15 times he can roll a 3 on one of the
dice. 14 of the times the dice will roll
randomly. The probability of winning on a
Hard 6 is the same as any random roller-- 1/11. The
15th time, however, he rolls a 3 on one of the dice. On that roll the probability of winning on a Hard
6 is ½. So the weighted probability P of
winning on a Hard 6 over this shooters 15 total throws is P = (14/15)*(1/11) +
(1/15)*(1/2) = 11.818%. Given this
probability, the shooter has an 18.2% edge over the house.
The shooter in this example was good enough to gain a substantial edge over the house by guaranteeing a 3 once every fifteen throws. A shooter can afford to be less frequent in rolling his target number yet still break even with the house. The shooter who targets a 3 can obtain 0% edge with the house if his skill guarantees him a 3 once every 45 throws. For this less-skilled shooter,
P = (44/45)*(1/11) +
When a shooter can win 10% of the time on the Hard 6 at a table that pays 9-to-1, then he is breaking even.
Below is a table of the target number, the optimal betting strategy, and the number of times, j, a player must roll the dice to guarantee hitting the target number:
So whereas a shooter who aims for a 2 on one of the dice would need to control his die 1 out of 28 times, another shooter aiming for a 4 would only have to control the die 1 out of 45 times. A 61% difference!
Calling Out the Braggart (The Chi-Squared Test)
A shooter boasts, I can control a die 1 out of 45 times, no problem. Ill make sure that when I roll, a 3 will appear more than usual. Well use 1/45 as my benchmark. He then proceeds to roll a die 300 times, scoring his target 60 times. Ha! Any random chicken feeder would have seen 50 3s. I rolled 60! Thats 20% more than expected. I beat my benchmark. Im a skilled shooter!
Is this shooter skilled? Should we put our money on the Hard 6 whenever he comes to the table? The Hard 6 is a risky bet, so wed better be sure that hes right. Skeptics would make him gather more data; believers would take the data as proof. What would statisticians do?
To determine whether or not this shooter can roll a 3 significantly more than a random roller, we need to treat his data set of 300 throws as points along a distribution and compare them to a uniform distribution of the numbers 1 through 6. The test to determine whether or not a distribution is significantly different than uniform is called the Chi-Squared Test. The test works like this:
-Make a hypothesis that says the shooter rolls 3s no different than a random roller. This hypothesis is called the null hypothesis.
To start, our null hypothesis is the following: Based on the 300 rolls of the shooter, he has rolled a uniform amount of 3s. The boasting shooter would disagree with the null hypothesis. But thats all rightthats just how these hypotheses are formed. In the end, the stats will determine the hypothesis veracity.
Chi-squared, denoted by X^2 (not to be confused with the letter x) is a common statistic used in discrete variable distributions. Just as a normal distribution has a Z statistic, the Chi-Squared Distribution has its own, namely X^2. One need not find X and then square it. X^2 is a telling number in itself and can be calculated from the following formula:
The braggart rolled the following:
It is not necessary to differentiate the total amount of, say, 4s from the total amount of 5s during this test. Later we will evaluate the uniformity of all the numbers, but now we are just concerned with whether the number he rolled was a 3 or not. We will treat 3s as a success and anything but a 3 as a failure. This can be described as a binomial distribution, since there are only two outcomes: success or failure.
For our binomial distribution, we need to determine the values of E(# of 3s) and E(# of non-3s). Naturally, the expected value of 3s for a random shooter is just 1/6 of the total rolls. In this case, 1/6 is 50. E(# of non-3s) = 250, i.e. Total Rolls - # of 3s.
O(# of 3s) = 60 as given by his data
set, and O(# of non-3s) = 240.
After we find the X^2 value, we use a X^2 table to look up the critical value. The X^2 table requires that we know the degrees of freedom and confidence level of the test. Without going into too much detail about the calculation of the degrees of freedom (denoted d.f.), we know that if we know the total number of dice thrown and the number of 3s thrown, then we can determine the amount of non-3s thrown. Similarly, if we know the total number of dice thrown, and the number of non-3s thrown, we can determine the total of 3s. This means there is 1 degree of freedom. (df = 1).
The confidence level of the test is an arbitrary number, but in most statistical forums, 95% confidence is the norm. We will use 95% to test the braggarts claim.
Using 1 degree of freedom and a chi-squared value of 2.4, the p-value is just over 10%. That is to say, there is more than a 10% chance of the null hypothesis is correct. In most forums, 5% is labeled statistically significant. Since the data suggests a higher percentile, we cannot reject the null hypothesis..
In other words, the braggart has not sufficiently shown that his roll was any different than a random roller. His better-than-average performance might have been a result of skill, but it might have also been a result of luck.
How large is large enough?
The goal in this section is to determine the expected number of rolls it would take to prove a shooter was capable of controlling a single die 1 out of 45 times. We saw in the example of the braggart that 300 rolls was too small a sample to prove this even with an impressive showing of 3s. We make use of the Chi-Squared table once again.
Start with the equation for X^2:
Lets let 3 be our target die again. Out of a sample size N, the expected number of 3s is given by N/6. The expected number of non-3s is 5N/6.
E(# of 3s) = N/6
Our observed data should reflect the fact that 1 out of 45 times, a 3 is guaranteed. Of the remaining 44 times, the roll is completely random. That is to say, if we have N rolls, 1/45 of them should be 3s and 44/45 of them are random (which also may turn up a 3).
O(# of 3s) =
(1/45)*N + (44/45)*(N/6)
Substituting our Es and Os,
On the Chi-Squared table, we are looking for the critical point at which we can be 95% certain the data suggest the null hypothesis is rejected. This step is as simple as looking at the Chi-Squared table where the p-value of .05 meets the d.f. value of 1. This intersection is given by X^2=3.85. Thus,
X^2 = 3.85 = N/405, and
Thus, a precision-shooter, skilled insofar as that he could guarantee rolling the 3 on every 45th throw, would be 95% certain that he was influencing the dice after 1,867 throws and no sooner. If a shooter didnt have the time to throw 1,867 trials but still wanted to test for non-randomness, he would have to compromise on the certainty of the data. For example, if he were willing to be only 90% confident in his ability to influence the dice as opposed to 95%, then N = 1314.
Testing more claims
from The Braggart
After showing the data to the braggart, he becomes frustrated. He is so confident in his precision shooting that he decides to make another argument. Clearly I have influenced the dice. Ive rolled sixty 3s and only forty-three 6s. I think that youve cheated me by considering my rolls to follow a binomial distribution. Lets just say that I throw a large amount of 3s and a low amount of 6s...better than the random roller at least.
A new challenge from the braggart! We can no longer treat 3s as successes and every other number as failures. He has stated, in other words, that he throws a non-uniform distribution of 3s and 6s. Fortunately, the chi-squared test applies here as well.
So first, we state the null hypothesis: The shooter throws a uniform amount of 3s and 6s.
Lets find the expected and observed values needed and then determine the chi-squared statistic for this hypothesis.
E(# of 3s) = 50
Turning to the Chi-Squared table, we find what percentile 3.8 falls under using 2 degrees of freedom. (We use two degrees of freedom here since there are 3 variables: # of 3s, # of 6s, and Total # of rolls). The table shows that p=15%.
Once again, we cannot reject the null hypothesis. In other words, we cannot say with 95% confidence that the shooter throws an unusual amount of 3s and 6s.
If you had a skill where you could roll a die and land it on one number more than the rest, you might think about going to a casino and cashing in on your gift. To optimize your profit, you would choose to target either a 3 or a 4. If you rolled the 3, you would play the Hard 6 and if you rolled the 4, you would play the Hard 8
To break even with the house, you would have to control the die 1 out of 45 times.
Assuming you could control the die 1 out of 45 times in a sample of at least 1867 throws, then you would be 95% sure you are skilled and not lucky.
But beware the braggart. Small sample sizes and hot streaks are enticing, but they are often misleading. If you followed his advice based off of a small data set, you would be playing a very dangerous bet. Remember that for a random roller, the Hard 6 and the Hard 8 are terrible bets the edge is 9% for the player. Youd want to make quite certain that he was more skilled at hitting the 3 or the 4 than the random roller. Use the chi-squared test for uniformity to test claims of precision shooting.
Mad Professor's Mini-Table Craps Tour with the Vegas Ghost- Part XI
dont get to play at this next venue very often.
In fact, the last time I was there was when I was picking up a comped
weeks use of a Shelby-American Series 1 automobile, courtesy of the fine folks at
absence didnt have anything to do with 9-11. Rather,
it was related more to the fact that the Speedway is a bit outside of my normal LV travel
range. There arent any other casinos
(with craps) within the immediate area, so it takes a dedicated trip to a decidedly
non-compelling location to get there.
Where Is It?
It is located way up at the northern end of the Las Vegas
Valley in a lower middle class suburb. To get
there, the quickest route is to take I-15 North to the Cheyenne Avenue exit and go east
for one block. You wont have a hard
time finding it once you turn left on Civic
not big or palatial, but from the parking lot to the craps table, it takes about one
hundred footsteps. This place is small,
friendly and never hosts any of the typical wall-to-wall tourist crowds that you see with
disturbing frequency on the Strip. Even
during the busiest times, there are never more than 200 people in the whole casino, and
that INCLUDES the entire staff. Like
I said, this IS NOT your typical tourist-joint.
Let me start by saying that the Casino is NOT associated with the
actual LV Motor Speedway, but they have borrowed the racing theme and junk
package that decorates the walls and ceiling as though they were.
place started out as the Cheyenne
Hotel & Casino, back in 1992 and switched over to the Speedway-theme in 1999 when it
became apparent that the 4-racetracks-in-1 LVMS was going to be a roaring success and draw
~140,000 patrons to this neck of the Joshua and Mesquite-tree woods on a regular basis.
Speedway Casino is part of a 95-room Ramada Inn that is attached to it. While there is really nothing memorable about
the casino, other than its small size, the mini-craps table is an entirely different
there is only one, that is indeed a mini-tub, and it offers some very fine
Precision-Shooting opportunities. It
comfortably accommodates 10 people (if they all shower on a regular basis, and 6 to 8
people if one or more of them dont).
first glance, the table seems to be bouncy in an uncushioned sort of way. The liveliness is caused by the use of ultra-thin
pure polyester felt. This is NOT the new-age
microfiber felt that we discussed in detail in my
Conquering Micro-Fiber Table-Felt article. Rather, it is just a cheap, thin layout over a
5/4-inch plywood base. With a
high-trajectory, high-backspin throw, the dice can sometimes leap more than a foot or two
in the air. Ive seen many new players
(new to this table) watch with alarm as their regular toss rebounds to the sky
like a ricocheting bullet. A
lower-trajectory, lower-energy, lower-backspin throw tames this green-felt beast in pretty
are almost always set at $2 with a $200 max-bet. Ive
never had the courage (read: stupidity and greed) to max-out my bets. The Pit Monkeys start to fidget like an
over-amped crack-whore if your bets get much beyond the $50 or $60 level for more than a
regular players are made up of locals whose income is on the lower side of the I
dont think we can really afford to be in here, but lets gamble anyway
scale. However, this being America, everyone
is entitled to put themselves as far below the poverty line as they wish, and the Speedway
seems like as good of a place as any for them to do it.
nice thing is if you go mid-month, the table is almost always sparsely-populated. This has held true since I started playing here
about a decade ago. In the ensuing time,
they havent always had a craps table in operation, but the current owners (silently
backed by gambling-investor and former Cheyenne owner Shawn Scott) feel that it is
important to maintain live-gaming so it doesnt take on the look and feel of a
does tend to get quite a bit busier, and a lot smokier at the end of the month and for the
first few days of a new month. Unfortunately,
the local gambling-wealth doesnt last much beyond a week or two around these parts. For the balance of the month, the table is
semi-populated with blue-collar guys, spending blue-collar money in hopes of turning their
blue-collar wages into money that would make Robin Leach eagerly send over a film-crew.
98% of Gamblers Lose
It doesnt matter whether your craps play is lit by the radiance of a Dale Chihuly chandelier at Bellagio, or the eerie nicotine-stained fluorescent glow at the El Cortez, 98% of ALL casino-players will continually lose.
task, is to ensure that your are in the OTHER 2% minority of whose who DO WIN
CONSISTENTLY. To that end, I would invite you to take a look at
my current 10-part series,
D'ya Wanna Win, or D'ya Wanna Gamble?
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