Let's take a fresh statistical look at dice control, starting with the basic physics principles described last month.

First and foremost, dice setters are trying to limit the dice rotation to the z-axis. This simply means that neither of the outside numbers shows up in the outcome. Hence, there are 4 times 4 = 16 possible outcomes that succeed in rotating in the Z-axis only. With a hardway set, this means each die outcome is one of: 2, 3, 4 and 5.

Secondly, a good dice setter tries to control the number of face rotations or pitches between the two dice when they do achieve z-axis control. Ideally, the two dice leave their hands in perfect symmetry, and rotate equally in the z-axis, bounce off the floor and back, and land with one of four possible outcomes. With a hardway set, this means one of: 22, 33, 44, or 55.

Thirdly, dice setters attempt to try to limit the number of double pitches when they achieve z-axis control. Ideally, the dice land up together (zero face rotations as noted above), or single pitch (are offset by 90 degrees or one rotation). Eliminating double pitches is useful with some dice sets, such as the hardway set: the only seven sums on the z-axis controlled outcomes occur when the dice double pitch (52, 25, 34 and 43). This can be relevant in different points of the game, such as point cycle rolls.

Thinking in practical terms, how do we know if a throw has stayed on z-axis rotation for any given dice set? Or, how do we know how many face rotations or pitches occurred on those successes? While it is very clear with a hardway dice set, other sets will lead to ambiguous results unless the dice are clearly marked (or colored) and the starting dice set recorded. For the purposes of the Pro Test©, we always use a hardway set, to eliminate this challenge. Hence, the actual dice sums are completely irrelevant, and say nothing immediately about our edge in craps. But we will come back to this point next month, and show later how we can determine our edge once we have proven our dice setting skill.

A random shooter should achieve 16 z-axis controlled outcomes on average for any 36 rolls. Of these passes, four are zero pitches, and four are double pitches. For the curious, observe that three pitches is actually a single pitch in the other direction. With this knowledge we can now ask the million dollar statistical question:

This fancy looking question is really the mathematical equivalent of asking "Is there ample evidence that I am a controlled shooter?"

First, let's call a z-axis controlled throw a Pro 1 Test pass. Hence, with a hardway dice set, a throw is a Pro 1 pass if the one and six don't show up in the outcome, and is a Pro 1 failure otherwise. Now, we can complete a roll set of throws, and record how many times we passed and failed the Pro 1 test (each outcome is either a pass or failure).

Now, if we really are capable of influencing the dice outcomes, then we should be able to record a bunch of throws, and the number of Pro 1 passes should be high enough to provide ample evidence of our skill. But how many passes in how many rolls is enough?

To answer these kinds of questions, mathematicians turn to statistical confidence measures. A mathematician will ask:

For our purposes, we'll say that if this percentage is less than or equal to 1%, then there is convincing evidence that our shooter is indeed influencing the dice. This means that only 1 in 100 random shooters would achieve that number of Pro 1 passes (or more).

This all sounds very convincing, but how do we convert a number of rolls and number of Pro 1 passes into a mathematically accurate confidence percentage or score? For this, we turn to a tried and true statistical method, called the Bernoulli trial.

Consider a statistical test (independent of craps) as follows:

• You can perform a test (event), which has either a pass or fail outcome. The fail outcome has probability p, and the pass outcome has probability 1 - p.
• You repeat this test n times, observing k failed outcomes, and n - k passes.
• Question: What is the probability of k (or less) failures?

This kind of statistical test is called a bernoulli trial, and is common fare in any first-year university statistics book or course. Here is the equation for the probability of failing k (or less) events (with probability p of failure each) in n trials:

The Bernoulli formula looks simple, but is not the kind of thing that can be computed by hand easily for any sizable 'n' (such as 100 or 500). But it is an easy matter to code this formula on a computer, which is exactly what is done in Smart Craps. When the program displays a Pro Test score, it is simply using the formula above to compute the statistical probability of the observed number of Pro Test passes (or equivalently, failures). When it displays the minimum number of passes required to pass a Pro Test in some number of rolls, it is simply running the Bernoulli equation a bunch of times to determine the number of passes that yields no more than 1% probability.

How does this relate to craps? Suppose a controlled shooter makes n throws with a hardway set, and we see k Pro 1 failures (i.e. n - k Pro 1 passes). Now, we can ask what the probability is of this occurring randomly using truly random dice. If this probability is sufficiently low, then there is a much greater likelihood that the shooter is instead successfully controlling the dice throws than the result being coincidence.

First, we need to determine p, the probability of a six or one (or both) showing up when using a hardway set (i.e. sixes and ones on the outside). The probability of a six or one not showing up on two independent and random dice rolls is 4/6 * 4/6 = 2/3 * 2/3 = 4/9. Therefore, the probability of seeing a six or one show up on either (or both) dice is 1 - 4/9 = 5/9. Hence, p equals 0.5555556 for our craps statistical test.

With this knowledge, we can solve the Bernoulli equation for any particular shooters' Pro 1 test score (probability).

The Bernoulli equation is very easy to determine from scratch. For those with a mathematical bent, here is the proof (and for those without, please feel free to skip this section):

• Suppose we ask what the probability is of exactly k failures in n trials with probability of failure p.
• There are (n choose k) possible permutations where we can have exactly k failures, and n - k passes.
• The probability for any one of these events is
• We can sum the probability of all these events to get the total probability of any exact k failures combination:

• Hence, the probability of k or less failures in n trials with probability p of failure is simply the sum of the above equations from zero to k (inclusive):

It should be noted that even on a computer, the Bernoulli equation is problematic for n (number of rolls) over 1000. In these cases, the first combinatorial term can quickly overflow the available precision for standard computer math libraries.

To solve this case, Smart Craps uses a standard normal approximation for the probability function when more than 1000 rolls are specified. While not as accurate as the above method, it is always within one roll of the actual exact result (with the Bernoulli equation). Given a desired probability (99%, or 1% by chance), we can solve for the minimum number of Pro Test passes:

MP = Minimum number of passes required
N(.99) = Normal cumulative distribution function for 99% (N(.99) = 2.326)

The second degree of freedom (or DOF) dice setters attempt to control is the number of face rotations or pitches on their Pro 1 passes (Pro 1 passes, or z-axis control, is the first degree of freedom). Let's call any Pro 1 Test pass that lands with both the dice in sync a Pro 2 Test pass. For example with the hardway set, the results 22, 33, 44 and 55 are Pro 2 passes, while all other results are Pro 2 failures. Notice that all Pro 2 passes are, by default, Pro 1 passes.

Now, we can ask the same question as before:

On the surface, it would seem that we just use the Bernoulli equation again, using [1 - 4/36] = 8/9 for 'p', with 'k' Pro 2 failures in 'n' rolls. There are 4 Pro 2 passes in 36 throws, which means 8/9 probability of a Pro 2 failure.

But statistical tests must be based on a solid knowledge of what they are testing, and how the variables relate. In this case, the hidden wrinkle is that all Pro 2 passes are also Pro 1 passes. Given the physics of dice control, this means all Pro 2 passes are dependent on the Pro 1 passes, which has great meaning in statistical terms. Statistically analyzing the Pro 2 passes without considering the Pro 1 passes, while possible, would lead to questionable results.

To see this, suppose a shooter manages to get 80 Pro 1 passes in 100 rolls, and 20 Pro 2 passes. 20 Pro 2 passes is much more than the statistical average of 4/36 * 100 = 11 in 100 rolls. But, we should also see 4/16=1/4 of our Pro 1 passes be our Pro 2 passes. So, 20 Pro 2 passes in 80 Pro 1 passes is actually statistically average. While showing good z-axis rotation control, the shooter is not managing to get zero pitches more often than expected. Analyzing Pro 2 passes compared to the number of rolls is misleading.

As such, the right statistical question to ask is:

From this point of view, 'p' is the expected number of Pro 2 failures ('k') in 'n' Pro 1 passes. There are 16 expected Pro 1 passes, and 4 Pro 2 passes. This means the probability of a Pro 2 failure ('p') is 1 - 4/16 = 3/4 (75%). Using the Bernoulli equation, we can then come up with the Pro 2 score or probability.

The third degree of freedom dice setters try to minimize is the number of double pitches. A double pitch occurs on a Pro 1 pass where the two dice are rotated by two faces in the outcome. Let's call a double pitch a Pro 3 Test failure. Unlike Pro 1 and Pro 2, we measure failures, rather than passes, with the Pro 3 Test. Hence, dice setters want to maximize their Pro 1 and 2 results, while minimizing their Pro 3 failures. Like Pro 2, all Pro 3 failures are by default Pro 1 passes. The hardway dice set, for example, has these four Pro 3 failures: 25, 52, 34, 43.

In the same way as we did with Pro 2, we can assess the statistical probability of some observed number of Pro 3 failures, using the Bernoulli equation. There are 16 expected Pro 1 passes, and 4 Pro 3 failures. This means the 'p' is 4/16 = 1/4, 'k' is the number of Pro 3 failures, and 'n' is the number of Pro 1 passes.

So how many test rolls do we need to pass the three Pro Tests? Using the Pro test Solver in Smart Craps, we can easily determine the requirements:

We are assuming the same 1% pass requirements in the table above as before. In 100 rolls, we only need 13 'extra' Pro 1 passes to yield a 1% pass, 18 in 200, and 21 in 300, etc. From the ratio column, we can see that the challenge of passing the Pro 1 Test gets easier as we add additional rolls. This is not surprising, since the statistical meaning of the results improves as we add more 'events' or rolls. The ratio of Pro 1 passes to number of rolls will approach the expected average of 44.44% (4/9) as the number of rolls increases. It is also equally true that a more skilled dice setter will be able to pass the Pro Test in less rolls than a modest performing shooter.

In tests with actual skilled dice setters, skilled shooters are capable of passing the Pro 1 Test in 100 to 500 rolls, in controlled test conditions. Measuring actual throws in a live casino is not advised, since the shooter most likely will employ dice sets that make recording Pro Test passes difficult. In all Pro Tests, the shooter must use the hardway set, and should record the actual throw as well as the Pro Test pass/failures.

Let's look at the Pro 2 pass requirements:

We used the minimum Pro 1 pass values as the basis above. Similar to the Pro 1 results, notice that the ratio of Pro 2 passes to Pro 1 passes decreases with increasing sample size, making the Pro 2 Test easier to pass as you add more rolls. The ratio of Pro 2 passes to Pro 1 passes will approach the expected average of 25% (1/4) as the number of rolls (and hence, Pro 1 passes) increases.

What happens if a shooter can pass one Pro Test, but not the others? This is completely acceptable with Pro Test, and Smart Craps allows you to test this. Let's look at different Pro 2 Test results in exactly 200 rolls:

Since the Pro 2 pass criterion is dependent on the Pro 1 passes, we need far less Pro 2 passes with low Pro 1 pass results. This may indicate a shooter who is not yet controlling z-axis rotation all that well, but when they do, they manage to keep the dice spinning together very well. In Smart Craps, you can enable and disable any combination of the three Pro tests.

Lastly, let's look at the Pro 3 pass requirements:

As with the Pro 2 results, notice that the ratio of Pro 3 passes to Pro 1 passes increases with increasing sample size, making the Pro 3 Test easier to pass as you add more rolls. The ratio of Pro 3 passes to Pro 1 passes will approach the expected average of 25% (1/4) as the number of rolls (and hence, Pro 1 passes) increases.

On the surface, mastering dice control seems easy… just roll the dice a few hundred times, and all you have to do is get about 1/2 of them or more as Pro 1 passes! What can be easier? Well, about 99 out of 100 things…

Remember that the passing metrics for all the Pro Tests is that there is 1% chance or less that a random shooter could replicate the result. This means it is really tough to pass! While looking deceptively easy on the surface, the reality, despite the attractive low pass numbers, is that it is very hard. It takes an incredible amount of practice and time to master the physical skill of dice setting, and even more effort to transfer that skill onto the casino floor and play with an advantage. Learning this skill cannot be done simply by reading books, like blackjack card counting: it takes actual practice throwing the dice, and lots of it!

So what if you fail the Pro Test? Does this mean you are not influencing the dice?

In a word… NO. Failing Pro Test does not mean you can infer the opposite result, that you are not a skilled dice controller.

Pro Test is a one-way test: it only provides statistical certainty that you are influencing dice outcomes. Due to the very high 1% confidence interval requirement for passing, failing, on its own, says very little about your lack of skill. Let's look at some examples.

Suppose you manage to roll 30 Pro 1 passes in 50 rolls. The expected number of passes was 22, so it sure looks like you are throwing in the right direction! But the actual Pro 1 Test score is 1.95%, so you failed the Pro 1 Test (you need 1% or less). But this does not mean that you should hang up your dice and find a new game! 2% as a Pro 1 score is very good, especially in 50 rolls (remember… the ratio of passes to rolls gets easier as you add more throws). Maintaining the same approximate pass ratio in more rolls is usually enough to manage a pass.

Suppose you throw an additional 50 rolls, and get 28 passes. This, by itself, is also a Pro 1 Test failure. But is we combine the results, we get 58 Pro 1 passes in 100 rolls. This is a passing Pro 1 Test result, with a score of 0.4418%… a very healthy pass! There is clear evidence that you are influencing the dice outcomes, in only 100 rolls.

Now, let's look at another example. Suppose you roll 100 rolls, and get 50 passes. The expected number for a random shooter is 44, and the Pro 1 Test score is 15.45%, clearly not a pass. But if we managed to continue with a 50% pass ratio, how many rolls would we need to pass the Pro 1 Test? If we get 250 Pro 1 passes in 500 rolls, the Pro 1 score is 0.7168%, a healthy pass (in fact we pass with 249 or more Pro 1 passes). So, there is good reason to continue and add more throws to the test, to see if you can pass.

But what if you fail miserably, say, 45 passes in 100 throws (a Pro 1 score of 49.4%, which basically indicates a random roll set)? Well, there are no indicators that adding more rolls will help you pass. But what if you simply weren't shooting very well during that test, and your physical skill was not working at its best? Dice control is a tricky talent much like any other skill requiring practice and effort. Dice setters often refer to being 'on' or 'in a rhythm', which is not necessarily voodoo and black magic. You may get good rolls one day, and not the next.

So, suppose the next day the same shooter above tries another 100 rolls, and does get 59 passes. Alone, this is a fantastic Pro 1 pass (0.2408%). Does this mean the shooter is clearly influencing the dice, and ready to make a fortune from the casino?

There is great temptation to exclude recorded roll sets arbitrarily. But if you did so, what would prevent a truly random roller from recording 100 or more 100 throw sessions, until they eventually pass (there is a 1 in 100 chance of a random shooter passing any of the Pro Tests)? For this reason, it is very important to aggregate your recorded Pro Test results, if you want meaningful measures of your skill. In the above case, the shooter actually had 104 Pro 1 passes in 200 rolls, which is Pro 1 score of 1.9069%. As with our first example, this is supporting evidence that the shooter is influencing the dice, but more rolls are necessary. They should practice some more, and roll an additional 100 rolls the next day when they are feeling 'strong'. Suppose they get 52 passes in the next 100 rolls, yielding 156 passes in 300 rolls. This is a Pro 1 score 0.5122%, a clear pass! As with most advantage gambling strategies, diligence, self-control and practice will help you become a better player.

Even extremely proficient dice controllers will find that it is very difficult to pass all three Pro Tests. From actual experimental results, those who pass tend to only pass Pro 1, or pass Pro 1 and only one of the other two. Also, Pro 2 tends to be the hardest test to pass, especially if you pass Pro 1, since you need all the more Pro2 passes (and less Pro 3 failures). For modestly skilled shooters, passing Pro 3 in isolation is the most likely case. Passing even one of the Pro Tests can often be sufficient to get a positive edge on some bets.

You've read the books about dice setting, practiced your throws, maybe taken a course, and now you're ready to see if you're just a chicken feeder (a random shooter), or hot dice controller. Pro Test is clearly the way to answer the question, but how do you go about doing it? Follow these general guidelines for taking the Pro Test:

• Learn the technique and skill: there is no point taking the Pro Test unless you have learned the fundamentals, and practiced. Dice setting is not a skill you can simply experiment with for a few minutes and take a stab at. Passing a Pro test in those conditions is a reflection of luck, and nothing else!
• Practice until it feels right: Do sports stars just hop on the court without warming up? Of course not… and it is no difference for dice setters. From a statistical point of view, it is perfectly acceptable to practice throwing until you feel comfortable and ready to take the test. If you just don't feel 'on', then wait until the next day.
• Get a good test environment: You might think that the best possible conditions for the Pro Test would be a live casino, with all the distractions and challenges of a normal craps table. This is not only impractical, but also incorrect. First off, the Pro Test requires a hardway dice set, so that we can easily determine the Z-axis rotations from the throw result. This is impossible with any different set, such as the 3V, and dice setters will (and should, to maximize their advantage), use different dice sets at different times in a real game. Secondly, the purposes of Pro Test is only to see if a shooter can influence the dice (converting the measured influence to an actual dice edge is covered later). Given the very high 1% confidence interval of Pro Test, it is far better to use ideal experimental conditions. Use of a throwing station in a quiet room is acceptable, as long as you are hitting the back wall most of the time (as one does in a real casino), and the station is similar to a live casino. Furthermore, your test environment (or station) may be worse than an actual craps tables, due to factors such as a bouncy floor on non-level height. So, make sure you give yourself the best possible conditions to throw the dice. Once you pass the Pro Test, Smart Craps uses, by default, a reducing factor (95% confidence interval) on your actual passing ratios to replicate the added challenge of a live casino.
• Get ready to record your throws: Smart Craps includes different recording sheets (Excel, and HTML web browser files) for you to use to record your throws. Stopping after every throw to record the result can be cumbersome and interfere with your ability to pass, so you might want to get a friend to record the throws. Either way, make sure you have an easy way to record your throws accurately.
• Start the test, and record every throw: Decide in advance how many throws you are going to record. You will be biasing the results if you instead simply start recording throws, and stop once you suspect your skill is fading. Choose a low number of rolls, such as 50 or 100, which will not tire you out. Once you start recording rolls, do not exclude any rolls! Doing so would only bias the results, potentially giving you false confidence from the scores.
• Aggregate your prior roll sets: While it may prove helpful to analyze the roll set data alone to determine 'local' trends, you should always sum together your prior recorded roll sets when looking for complete validation of your skill. Long run, you will get generally better Pro Test passing scores as you add more rolls, assuming you are influencing the dice (remember: the Pro Test gets easier to pass as you add more rolls. This is the nature of statistical certainty tests). There are times though when you can safely start excluding roll sets (from a statistical point of view), especially over the long run. For example, your dice setting skill may improve over time, in which case including old roll sets can exert overly negative influence on your scores. Removing all roll sets older than a certain time would be acceptable. Excluding a hodge-podge of roll sets you just didn't like would be unacceptable, and bias the scores.

Pro Test provides a statistically valid and solid foundation for determining whether a shooter who claims to have dice control skill is in fact influencing the outcomes. The Pro Tests have a 100% correlation to the common physical understanding of dice control (they maximize the information content of dice control, in mathematical terms). By modeling the statistical occurrences of Z-axis rotations, Pro Test is the most accurate possible measure of dice setting and influence. Within a few hundred rolls, good dice setters can state with statistical certainty that they are controlling the dice.

But what about player edge? Next month, we'll see that mathematical modeling of player edge is possible with Pro Test, once again with the help of computer code implemented in Smart Craps.