Hand Length – What Are True Odds? by Dan Pronovost
Posted by Dan Pronovost

With the new "40 and over record holder’s" club Frank and Dom have created, I’ve become very curious about the probability of such events (as a number of others have too). So, I’ve tuned up Smart Craps with a few extra features, and done a bunch of analysis. This should (hopefully!) answer questions about just how rare and/or difficult such things are.

Now… let’s start with the summary of the data, and get to the boring math stuff later…

Here’s the data for a random shooter:
50% of hands are of throw length 6.96 or more (median value)
5% of hands are of throw length 23.44 or more
1% of hands are of throw length 34.95 or more
0.49% of hands are of hand length 40 or more
0.12% of hands are of hand length 50 or more

From the above, we can conclude that inclusion of the ’40+ rolls’ club is indeed hard to do… about 1/2 of 1% of the hands (for a random shooter) will qualify. A random shooter would have to complete 141 hands to have a 50/50 chance of seeing a 40+ roll! Let’s assume maybe 2 or 3 hands an hour for a player on a regular night, and 4 hour sessions… a random shooter will see a 40+ roll only every 14 or so playing sessions! If you play once a month on average, that’s basically once a year.

So… the next question, is it much easier for a skilled dice controller to get into the 40+ Club? My initial guess was that it would be immensely easier, because I’ve already shown that the average hand size (different from the median) is one to three rolls higher than a random shooter. But exponential series are tricky… and statistics can be surprising!

Now, let’s look at an expert shooter (really good Pro Test scores, but not godly):
50% of hands are of throw length 7.30 or more (median value)
5% of hands are of throw length 24.92 or more
1% of hands are of throw length 37.23 or more
0.70% of hands are of hand length 40 or more
0.19% of hands are of hand length 50 or more

We see that while the expert will get longer hands more often, it’s not by very much. A fraction of a percent is still a fraction.

I also tested and studied different kinds of shooters, from godly, to godly on a hot streak. Even with godly skill on a hot streak (which means no confidence interval reduction of godly values), the 50+ hand probability was only 1.59%. A big jump (many times), but still not an easy task! 3.78% for the 40+ club for a super god, BTW.

For the really curious, I have six sim reports that you can look at that show various combos of skill and dice sets I tried out (I tried optimal pass line, all hardway, all-sevens/hardway combos). Here are the links:

random
expert Pro test
expert Pro Test shooter, hardway set always
expert Pro Test shooter, hardway set on points, all sevens on come out
Godly Pro Test shooter (real data… guess who! Starts with F, ends with S)
Godly Pro Test shooter, no confidence reduction, so super godly!

If you scroll down in the reports, you’ll find the

Now, for those who want ‘Distribution table of hands by hand length’ sections. It’s all there.

Now, for the boring math stuff (that proves and backs up the above, to quell the doubters)…

First, I added a ‘hand length distribution’ table in the latest Smart Craps (v1.42, now up for download). This shows the full distribution of hands by size across the simulation. BTW… I call the time from a shooter getting the dice to sevening out a ‘hand’.

I also added a statistical model of hand length to Smart Craps. In theory, hand length should follow an exponential ‘lifetime’ model. In short, you can approximate hand length with the equation: pr = p ^ (n – 2) (-2 because hands of length 0 and 1 are not possible). Here n is the hand length, p is the (inferred) probability of NOT getting an event that terminates the hand (i.e. not sevening out), and pr is the resulting probability of getting that hand length or more.

Now, people might intuitive think that p, for a random shooter, is 5/6 (i.e. not rolling a seven). But it isn’t, because there are hands where you might get 2, 3, 7, 11, or 12 on the come out roll, yet continue. Plus, hand lengths of 1 are impossible in craps. This is why computing the mathematical probability (for a random shooter), as Professor Catlin did in his excellent article, is so bloody complicated and hard! Tricky stats!

But, it occurred to me that the exponential model would probably be a very good approximation. And, we could use the distribution table to approximate p. With this variable solved, we can then solve for pr or n. And, I could compare the actual emprical results for distribution to the expoential model. From the links above, you can see in the tables that the two correspond perfectly. The exponential is a fine and fair approximation, for both random and skilled dice controllers.

Conclusion: if you get into the 40+ roll club, congratulations are deserved. It ain’t easy! Howard’s record is quite amazing, BTW, with the multi-day streak.

Dan Pronovost
www.smartcraps.com

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