Optimal Dice Set for Dice Controllers in Craps By Dan Pronovost
Introduction In my craps article in last month's BJI newsletter, I continued introducing readers to Smart Craps This month, we'll see that not only can we use Pro Test to determine if we are good dice controllers (instead of just lucky), but also to determine our actual - Craps article #1: www.bjinsider.com/newsletter_62_dice.shtml
- Craps article #2: www.bjinsider.com/newsletter_63_dice.shtml
Pro Test Player Edge So… you've practiced your dice control, recorded a few hundred throws, and pass some (or all) of the Pro Tests. Does this mean you're going to make a killing at the casino? What is your player edge (or expectation) on different bets? While not immediately obvious, it is possible to convert Pro Test results into game expectations. To do so, you need to know: - The rules of the craps game you are playing in, such as the odds and pay schedule.
- The specific bets you are going to make.
- The dice sets you will use at each point in the throwing cycle.
- Your Pro Test results.
2) Craps game simulation. Converting Pro Test results to exact player expectation Normally for a random shooter, each of the 36 dice outcomes has exactly 1 in 36 chance of occurring. For dice setters, the odds will vary for each potential outcome in a predictable manner. Once we have a combination of Pro Test results (a shooter may only pass one or two tests, or possibly all three), these can be converted into specific probabilities for each of the 36 outcomes. Once we have this and the dice sets used at every point in the game, we can determine the probabilities for each dice sum. And with this knowledge, we can determine the actual player edge given a specific betting pattern and game. We can do this mathematically, without empirical simulation. Suppose a shooter passes all three Pro Tests with results
For each group above, the outcomes in that group each have the same probability of occurring. For a random throw, this would be exactly 1 in 36. But for a controlled shooter, it will vary. If a shooter passes the Pro 1 test with a value of For a controlled shooter (say, 57 Pro 1 Passes in 100 rolls), the probability of Pro 1 failure is: 1 - (57/100) = 43/100 = 43.00%. This is significantly less than the random shooter (55.56%, above). If a shooter passes the Pro 2 test with a value of For a controlled shooter (say, 23 Pro 2 passes in 57 Pro 1 passes), the probability of all Pro 2 passes is 23/100 = 23.00%. This is significantly higher than the random shooter. If a shooter passes the Pro 3 test with a value of For a controlled shooter (say, 6 Pro 3 failures in 57 Pro 1 passes), the probability of all Pro 3 failures is 6/100 = 6.00%. This is significantly lower than the random shooter. Now, armed with this approach and a given dice set, we can exactly state the probability of each of the 36 outcomes for a controlled shooter. This is not a simple mathematical operation to complete by hand, by is easy to do with a computer and a bit of code (and this is what Smart Craps does). Then, we can simply sum the probabilities for each unique dice sum (2 to 12), telling us the probability of each roll in craps, for the specified dice set. If we look at a specific bet, such as a pass line bet, and specify the dice set for each situation (come out roll, 4/10 points, 5/9 points, 6/8 points), we can now come up with an actual player edge. For example, on the come out roll, we know that a 7 or 11 will pay 1 to 1, and 2, 3 and 12 will lose our bet. Each of these four sums will have an exact probability given a dice set and specified Pro test results, as shown above. Following this approach, it is possible to write a precise equation for the player edge, in terms of the probabilities for each dice sum given a set of specified dice sets. The mathematics is extremely complex and presented separately after this section, but thankfully you don't need to know it… So far the edge calculator sounds pretty good, as long as you know the dice sets that you use for each situation in the game (such as come out roll, and points). But the analysis method above could help us determine the - For each 'situation' (come out roll, 4/10 point, 5/9 point, 6/8 point), do the following:
- Test all 576 dice sets, and see which one contributes the greatest player edge (or least loss). This can be done using the same analysis approach as above, considering only the win and loss outcomes for the situation. For example, on the come out roll, the edge contribution is
*p7*(probability of a 7 sum) +*p11*-*p2*- p3 -*p12*. The remaining outcomes are points, and a similar (but more complicated) equation describes their contribution. - Of the 576 dice sets, one or more will be optimal, providing the greatest contribution.
- Take the best dice sets for each situation, and re-compute the actual player edge (given the Pro Test analysis approach noted previously).
Once again, we can see that computing optimal dice sets by hand is not possible. But a computer can do it trivially, which is what the Dice Set Optimizer does in Craps simulation The edge calculator and Dice Set Optimizer are great for simple bets and games, but the real world of craps is filled with twists and variation. What about unusual bets, such as vig or don't bets? What if you bet occasionally on random shooters? What if you vary the size of your bets according to a 'system'? While the edge calculators in This is where the Summary Pro Test, the Dice Set Optimizer, and the simulator in
For the technically minded folks out there, we also have a complete mathematical derivation and proof of the equations used to compute the player edge in The complete presentation on Pro Test, edge calculation, and mathematical formulas is freely available in PDF format at: www.smartcraps.com/SmartCraps_theory.pdf Near the end is the mathematical derivation and edge equations. This section provides a mathematical derivation and proof for how the edge calculator works in
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