SMARTCRAPS rtf Smart Craps Page 25 of 163, Figure 2 Throwing the dice parallel to the Z axis of the table. A typical controlled shooter is attempting to cause the dice to rotate. only around the Z axis In theory any spin in the Y or X axis can. introduce unexpected bounces and a less controlled outcome With a. spin in only the Z axis the outside numbers on the dice set should not. occur in the final roll result, Now that we understand the basic physics of dice control we can see. that the actual dice set is relevant for a skilled shooter The set. determines exactly what numbers are on top and on the sides for both. dice In the above figure the twos are on top and the fours are on the. front The standard notation used in Smart Craps would describe this. dice set as T2F4 T2F4 meaning the two on top and four in front for the. left die and right die The notation T3F5 T3F1 is often called the 3Vset. in popular texts for example Describing the top and front of a die. actually any two adjacent sides will do completely determines the. values of the remaining sides The T and F can be dropped. shortening the notation for a set such as the 3V to 35 31. As well as limiting the spin of the dice to the Z axis extremely proficient. controlled shooters also try and fix the rate of spin for both dice to be. the same This can further reduce the dice outcomes for advantage. SMARTCRAPS rtf Smart Craps Page 26 of 163, play For example with a 24 24 or hard way set sixes and ones on the. outside a perfectly controlled throw with Z axis rotation only at the. same speed for both dice means the outcome should be one of 22 33. Using a hard way set again suppose a controlled shooter achieves Z. axis control but not rotation rate In theory the possible outcomes are. the sixteen permutations of the four on axis sides 22 23 24 25 32. 33 34 35 42 43 44 45 52 53 54 and 55 Notice that there are. passing outcomes that sum to seven as well as failures 61 16 This. will come into play when we talk about the statistics of dice setting and. the best way to determine if a skilled shooter is influencing the. In the next sections we discuss two ways of determining if a shooter is. in fact influencing the dice outcomes SRR Seven to Rolls Ratio and. the new Pro Test by DeepNet Technologies,Next topic in this section. SMARTCRAPS rtf Smart Craps Page 27 of 163,SRR Seven to Rolls Ratio. Ok you claim to be an expert dice controller and making millions at. craps Good for you Now how do we prove it with the same kind of. mathematical certainty that we talk about with card counting in. The most common statistical analysis of the game of craps to date is. the SRR seven to rolls ratio This measures the number of seven. sums in two dice compared to the other outcomes If you are a random. shooter you will roll one seven in every six rolls or one seven for. every five non sevens This simple fact can be seen if we look at all. possible outcomes for the sum of two dice,Roll 2 3 4 5 6 7 8 9 10 11 12. 11 12 13 14 15 16 62 63 64 56 66,21 31 41 51 61 26 36 46 65. 22 23 24 25 35 45 55,32 42 52 53 54,Total 1 2 3 4 5 6 5 4 3 2 1 36. Figure 3 Dice outcomes for a random shooter, The above table for a dice controller will be quite different but the. number we concentrate on is the number of seven outcomes On the. come out roll we want to maximize the number of sevens i e have the. lowest possible SRR ratio and on any point roll we want to minimize. the number of sevens i e have the highest possible SRR ratio We. should note that minimize and maximize may seem backwards in the. above description technically you want to maximize the SRR fraction. on a come out roll for example which means making the second. number in the ratio proportion lower In general we refer to minimizing. the SRR to mean making the second number in the SRR ratio as low. as possible and maximizing the SRR to mean making the second. number in the SRR ratio as high as possible, In Smart Craps we model an SRR shooter by specifying both the. minimum and maximum SRR ratio they are capable of achieving. Then we can devise a betting system that uses the appropriate SRR. SMARTCRAPS rtf Smart Craps Page 28 of 163, for different situations the minimum SRR on come out rolls and the. maximum SRR on point rolls, With the above approach we can convert various SRR ratios to actual. player edges In Smart Craps we do this by defining a simulation. matching the SRR parameters and running it for a few million rounds. For example if we assume the following, Shooter uses a controlled throw on the come out roll that minimizes. their SRR to 1 5 5, Shooter uses a controlled throw on all point rolls that maximizes. their SRR to 1 6 5,Zero odds game with flat pass line bets only. Player edge 3 43 using 100 million craps rounds where each. round is a single bet 342 million rolls in this case. While it is possible to determine the player edge mathematically with. different SRR values in the case of simple flat pass line bets things get. more complicated when you add unusual bets and rules vigs odds. uneven payouts etc Smart Craps makes it possible to determine the. true edge empirically with real world craps scenarios and conditions. using game simulation, It is interesting to observe that the actual dice set is not relevant when. simulating SRR shooters The SRR is a direct result of their technique. and set not the other way around But SRR shooters are no different. from any dice setter the principles of axis rotation discussed previously. Also the actual dice sums while very relevant to computing player. edge are not directly correlated to the influence the dice setter. achieves For example if a shooter uses a hardway set they may still. achieve perfect z axis rotation not getting an outside six or one to. appear yet a seven is rolled For example the outcome 34 and 25. indicate perfect z axis rotation with a hardway set Yet they may fail to. get z axis rotation and still get a seven with a 61 or 16 outcome This. weakens the statistical accuracy of the SRR as a good measure of dice. setting skill SRR is great for computing player edge but is a weak. statistical measure of dice setting skill, Enter Pro Test a fresh look at the statistics of dice setting and a. better mathematical approach to measuring skill and player edge. Next topic in this section,SMARTCRAPS rtf Smart Craps Page 29 of 163. Pro Test a better dice control metric, We ve seen in the SRR Page 27 section that the seven to rolls ratio. while somewhat intuitive and useful for calculating player edge is not. the best possible measure of dice setting skill This results from the fact. that sevens can be both good and bad meaning that they occur. when we both achieve and fail z axis control This weakens the. statistical utility of SRR for determining player dice setting skill. Let s take a fresh statistical look at dice control starting with the basic. physics principles described in the Theory of Dice Control Page 24. Z axis rotation, 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 and show later how we can. determine our edge once we have proven our dice setting skill. SMARTCRAPS rtf Smart Craps Page 30 of 163, 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. If I roll the dice a bunch of times and observe a certain number. of z axis controlled shots what is the probability of this occurring. This fancy looking question is really the mathematical equivalent of. asking Is there ample evidence that I am a controlled shooter. Next topic in this section,SMARTCRAPS rtf Smart Craps Page 31 of 163. Pro 1 Test, 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, What is the probability of a random shooter achieving that many. Pro 1 passes or more in some number of throws, 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,Bernoulli trials. 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. 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,P p i 1 p n i,SMARTCRAPS rtf Smart Craps Page 32 of 163. Figure 4 Bernoulli equation, 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,Bernoulli equation from first principles. The Theory of Dice Control This can further reduce the dice outcomes for advantage SMARTCRAPS rtf Smart Craps Page 26 of 163 play For example with a 24 24 or hard way set sixes and ones on the outside a perfectly controlled throw with Z axis rotation only at the same speed for both dice means the outcome should be one of 22 33 44 or 55 Using a hard way set again suppose a

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