Hand Lengths in Craps:
How Impressive are Those Monster Rolls?
By Dan Pronovost
Dan Pronovost is the owner and president of DeepNet Technologies, makers of a wide range of gambling training products and software. Their web site are: www.HandheldBlackjack.com and www.SmartCraps.com and all products are available for free trial download. Dan is the creator of the new card counting system Speed Count, which is being taught by Henry Tamburin and Frank Scoblete in the Golden Touch Blackjack two day courses: www.GoldenTouchBlackjack.com.
This is a continuation of on-going articles on the game of craps and advantage dice control. As with prior articles, we are using new and powerful mathematical tools to analyze craps, now available in our software Smart Craps. This article refers to Pro Test and other statistical metrics described in our prior articles to date. If you haven't yet read these articles, you probably should to understand what follows:
- Craps article #1: www.bjinsider.com/newsletter_62_dice.shtml
- Craps article #2: www.bjinsider.com/newsletter_63_dice.shtml
- Craps article #3: www.bjinsider.com/newsletter_64_craps.shtml
Hand Length in Craps
Traditionally, there are two justifications dice controllers use to provide evidence of their skill and ability to make money playing craps: 1) personal win history, and 2) hand length or number of rolls.
The former argument usually goes along the lines of, "I've been a dice controller for years, and I've made a lot of money doing it. I win more than I lose." While personal fiscal success or failure with dice control can be very convincing for the person involved, it unfortunately turns out to be a very bad statistical indicator, meaning it doesn't prove much by itself. As with most advantage gambling strategies, such as card counting in blackjack, the potential edge with dice control is generally pretty slim (1% to 3%), which leads to large bankroll (i.e. profit) variability. A few sessions, or even 100 hours, of slight gains does not constitute statistically convincing evidence. And in fact, the reverse can be true: a person who is indeed exercising healthy dice control may simply get unlucky and still have losing sessions. As we have discussed in our prior articles, a much more accurate and meaningful measure of skill is Pro Test, which measures axial and rotational control over the dice, based on the physics and principles of dice control. This maximizes the statistical information, yielding a good test that can prove dice control influence in as few as 100 to 500 rolls for skilled shooters.
The second common argument for dice control skill is reference to long hand length in craps. Hand length refers to the number of dice rolls a shooter has before relinquishing the dice to the next player (after sevening out on a point roll). 'Monster rolls' of 40 or more are often achieved and documented. What can these occurrences tell us statistically about dice control skill?
Measuring Hand Length Probability
To understand if some occurrence(s) of hand length are rare or not, we need to find out what a random shooter would expect for number of rolls in craps (hand length). It is possible to determine expected hand length for a random shooter via pure mathematics, but it requires very complicated techniques. Professor Don Catlin, a well known math expert on gambling, wrote an excellent treatise on this exact subject in a Casino Times article: http://catlin.casinocitytimes.com/articles/1232.html
But this approach only works for random shooters, and we'll need to also test hand length for shooters with pre-supposed dice control skill. So, instead we'll use the craps simulator in Smart Craps to simulate many millions of dice throws in craps, and empirically see the resulting hand lengths. With this approach, we can test both random and non-random shooters.
We setup a craps simulation with the following parameters:
- 10,000,000 betting rounds (about 20 million dice throws).
- One pass line bet (only) per round.
- One random shooter.
The following distribution table shows some of the outcomes for hand length for a random shooter. This is a summary from a much larger table in the Smart Craps report file for the simulation (see http://www.deepnettech.com/smartcraps009.shtml for the complete report).
Hand length | Frequency | Probability for hand | Prob. hand >= |
1 | 0 | 0.00% | 100.00% |
2 | 263,892 | 11.11% | 100.00% |
6 | 187,113 | 7.88% | 57.60% |
7 | 162,738 | 6.85% | 49.72% |
20 | 23,661 | 1.00% | 7.28% |
30 | 5,441 | 0.23% | 1.66% |
40 | 1,215 | 0.05% | 0.38% |
50 | 288 | 0.01% | 0.09% |
In addition, the Smart Craps report includes the following special statistics:
- Average # of throws/hand: 8.53
- 50% of hands are of throw length 6.96 or more (median value)
The median value is 6.96, which means 50% of hands will be under 6.96 rolls inclusive, and 50% will be longer than 6.96 rolls. Mathematically, we call this the median value. Interestingly, the average number of rolls is much higher than the median value. This is not surprising in this case, as hand length in craps generally follows an exponential, or lifetime, probability distribution.
We can also see that long hands are fairly rare. We would expect hands of length 30 or more only 1.66% of the time, for example. This means that if we walked up and threw 100 sessions or hands of craps, we'd expect only one or two of these to be of length 30 or more. Hands of length 50 or more are very rare (about 1 in 1000).
Note that there were no hands of length 1 (one roll), since it is not possible with the rules of craps. At worst, you will roll a point number, and then a seven (two rolls). When you roll a 7, 11, 2, 3, or 12 on the come out roll, you still get to 'hold' the dice and continue throwing.
Hand length with skilled shooters
With our new understanding of hand length probability for random shooters, we can now take a closer look at skilled dice controllers. Once again, we setup a simulation in Smart Craps as above, but we used a skilled Pro Test shooter instead. The shooter has Pro Test settings representing extremely high level of dice control skill, resulting in a potential single odds pass line edge of about 8.5% (determined from the Smart Craps edge calculator, and confirmed with the simulator). Here are the settings we used in the simulation:
- 10,000,000 betting rounds (about 20 million dice throws).
- One pass line bet (only) per round.
- One expert Pro Test shooter: # rolls: 200, Pro 1: 106 passes (0.922904% Pro Test score), Pro 2: 38 passes (0.841179% Pro Test score), Pro 3: not used.
- Optimal dice sets on come out and point rolls, as determined by the Smart Craps dice set optimizer.
The following distribution table shows some of the outcomes for hand length for a very skilled dice controller. This is a summary from a much larger table in the Smart Craps report file for the simulation (see http://www.deepnettech.com/smartcraps010.shtml for the complete report).
Hand length | Frequency | Probability for hand | Prob. hand >= |
1 | 0 | 0.00% | 100.00% |
2 | 223,179 | 10.28% | 100.00% |
7 | 145,546 | 6.70% | 52.35% |
8 | 127,378 | 5.87% | 45.64% |
20 | 24,428 | 1.13% | 8.77% |
30 | 6,243 | 0.29% | 2.21% |
40 | 1,493 | 0.07% | 0.55% |
50 | 406 | 0.02% | 0.14% |
In addition, the Smart Craps report includes the following special statistics:
- Average # of throws/hand: 9.05
- 50% of hands are of throw length 7.30 or more (median value)
We can see that even with this extremely high level of dice control skill, randomness still dominates the game of craps. The median value has jumped up marginally to 7.3 rolls. The probability of a hand of length 40 or more went up from 0.38% to 0.55%. This is nearly a 45% increase over the random shooter, but in either case it's still pretty rare!
Conclusions
While monster 40+ rolls are very impressive and rare, on their own they provide very little evidence by themselves about the skill of the person achieving them. They are rare events for both random shooters, and expert level dice controllers. Also, given enough chances to hold and throw the dice, monster rolls are bound to happen. If we assume at most 100 throws an hour at a craps table, we can roughly estimate 10-12 hands an hour, regardless of the skill the shooters. Hence, a single craps pit should see a 40 or more monster roll about once a day.
So, monster rolls are going to be a daily event in any casino, regardless of the skill of the shooters.
The fact that roll length is not necessarily a good indicator of dice control skill may seem counter intuitive and even disappointing. How can it be that a known skill level, represented by a pure metric like Pro Test or the SRR, yield a positive edge for the craps player, yet their expected hand length is not all that different from random? Advantage gambling, including dice control, usually means only a slight shift in the player's favor, and such marginal edge factors mean that win/loss records and roll length, especially over short periods, are not compelling statistical evidence.
Monster-Monster rolls…
But what if a shooter claims to have a very high proportion of monster rolls (I'll call this monster-monster rolls)? Does this say anything statistically?
First, it is only human nature to see patterns where we want to, and ignore what doesn't fit. A proper assessment of hand length over time would require the player to record both when they achieve long rolls, and when they don't. Even a random shooter will occasionally have great monster rolls, as we've shown above. I mention this because we need to objectively assess how likely it is, even for a very skilled shooter, to get an extremely high proportion of monster rolls.
To test this, let's suppose an even higher level of dice control skill, what I'll call "Super Godly". These Pro Test metrics and simulation results are near super-human, and the best I've ever been provided. But it will help us test 'monster-monster' roll probability, even assuming the most amazing of skill levels. Here is the simulation I ran (see http://www.deepnettech.com/smartcraps016.shtml for the complete report):
- 10,000,000 betting rounds (about 20 million dice throws).
- One pass line bet (only) per round.
- One expert Pro Test shooter: # rolls: 412, Pro 1: 297 passes (0.00000% Pro Test score), Pro 2: not used, Pro 3: 41 (0.000161% Pro Test score).
- Optimal dice sets on come out and point rolls, as determined by the Smart Craps dice set optimizer.
- Pass line single odds potential edge: 21.371383% (with 95% confidence interval reduction on Pro Test results…see prior articles for more on this)
Now, with this godly dice control skill level, they still only have a 1.23% chance of having a 40 or higher roll (compared to 0.38% from above for a random shooter). Suppose the shooter claims that they have hit ten 40+ hands in the last 100 times they got the dice. Is this probable with their skill level? Using the same Bernoulli equation used for Pro Test (see prior articles), we can compute this probability. The probability that this godly shooter can achieve ten (or more) 40 or higher roll hands in 100 opportunities of holding the dice is… 0.0001%.
We've already assumed an incredibly high level of dice control skill, yet the likelihood that they had 10 (or more) 40+ hands in 100 sessions is one in 1 million! We're left with assuming that they have an even higher level of skill (unlikely), or that some poor hand lengths were not recorded.
Ultimately, the marginal difference in monster roll probability between random and skilled shooters makes this a weak statistic for providing evidence of dice control. Pro Test, since it statistically tests exactly the expected results from the physics of dice control, is the most accurate and useful measure of dice control skill.
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