tysen2k

The average is given previously in this thread (page 5). The whole purpose of limiting it to 8 cards was to test the baseline of point methods where we don't add anything for extra trumps. But sometimes we know for a fact that we do only have 8. How much should the 8-card suit be worth opposite a void? If partner has something, then we add more points.

Was Goren right after all? So I decided to look at the cases where we have exactly an 8-card fit. I looked at the trick-taking potential of each shape. Tricks Tricks Tricks Tricks Tricks Error Error Error Error Shape Count Real Zar 531 TSP Goren Zar 531 TSP Goren 4-3-3-3 10.54 0.00 -0.29 -0.22 -0.14 -0.08 0.90 0.49 0.20 0.08 4-4-3-2 21.55 0.22 0.11 0.12 0.06 0.25 0.30 0.25 0.56 0.01 5-3-3-2 15.52 0.23 0.31 0.12 0.26 0.25 0.10 0.19 0.02 0.01 5-4-2-2 10.58 0.43 0.51 0.45 0.46 0.58 0.07 0.01 0.01 0.25 6-3-2-2 5.64 0.46 0.71 0.45 0.66 0.58 0.33 0.00 0.22 0.08 6-3-3-1 3.45 0.65 0.91 0.78 0.86 0.58 0.22 0.06 0.15 0.02 5-4-3-1 12.93 0.66 0.71 0.78 0.66 0.58 0.03 0.20 0.00 0.08 4-4-4-1 2.99 0.68 0.31 0.78 0.46 0.58 0.41 0.03 0.13 0.03 7-2-2-2 0.51 0.82 0.91 0.78 1.06 0.92 0.00 0.00 0.03 0.00 6-4-2-1 4.70 0.85 1.11 1.12 1.06 0.92 0.30 0.33 0.21 0.02 5-5-2-1 3.17 0.91 0.91 1.12 1.06 0.92 0.00 0.14 0.08 0.00 7-3-2-1 1.88 0.91 1.31 1.12 1.26 0.92 0.30 0.08 0.24 0.00 5-4-4-0 1.24 1.16 0.91 1.45 1.06 0.92 0.08 0.11 0.01 0.07 5-5-3-0 0.90 1.16 1.11 1.45 1.26 0.92 0.00 0.07 0.01 0.05 6-4-3-0 1.33 1.19 1.31 1.45 1.26 0.92 0.02 0.09 0.01 0.10 6-5-1-1 0.71 1.29 1.31 1.78 1.66 1.25 0.00 0.17 0.10 0.00 6-5-2-0 0.65 1.47 1.51 1.78 1.66 1.25 0.00 0.06 0.02 0.03 Totals 3.06 2.29 2.00 0.83 Count is the % frequency of that shape Tricks Real are the number of tricks that each shape really takes more than the 4333 shape Tricks for each evaluation scheme are the number of tricks predicted by that count system. I allowed everything to be shifted by a constant so that you won't have a problem if the 4333 shape is off. This helps Zar's performace by a lot. It would be worse without it. Error is the square of the difference between the real and predicted numbers, multiplied by the Count. Total at the bottom is the sum of the errors. It looks like most of the systems are predicting more tricks than are really available. Good old Goren is the closest! Implications? Distribution when we only have an 8-card fit maybe doesn't have as much weight as a lot of us were thinking. But since it has much more weight on average over all possible fits, that must mean that we really have to increase it a lot when we have a superfit. So maybe the best solution is for all methods to tone down a bit closer to Goren as the baseline, but once a superfit is found, pump it up even more than Zar or TSP ever did before? I'd like to hear other people's thoughts.

Also, in order to look at adjustments, you have to have a baseline right? I'm going to do this anyway so that we can compare baselines. For each evaluation scheme, the baseline has always been set as its value in an 8-card fit.

Did you read the two long articles I referred to earlier on adjustments?

Naturally they are very important in determining both the offensive and defensive potential of our hand.

This line of study has also made me to start thinking about what hands make the best preempts. Here's something I was thinking about and let me know if you think it's the right track. If you look at the par contract for a DD deal, there is some chance that the par contract is a sacrifice for us. I'll try if I can find some characteristics that would predict the chance that par is going to be a sacrifice for us. Would that be a good indication of whether it is likely to be a good preempting hand? I think it would combine all the needed elements: Enough trick taking ability for a sac to be profitable Our opponents having a good enough hand on their own that we need to sac Not having a majority of the strength since otherwise they would have the sac against us Is it clear what I'm proposing? Note that it would entail not only the estimated offensive and deffensive strengths of our hand, but the exact hand pattern as well, not just majors vs. minors. So it would get pretty complicated. I don't know if anything meaningful would come out of it. What do you think?

Okay, I think I have an idea of how to solve the "distribution accuracy" question and get over the whole upgrading/downgrading issue. I haven't run this yet since I'm at home now and all of my bridge stuff is on my work laptop. Ben, let me know if you think this is a fair test. We'll look at the average number of tricks that each shape takes like we did before, except that I'll limit the hands to be only those where our longest fit is exactly 8 cards. Since every distribution method counts an 8-card fit as "normal" with no adjustments, it should be fair to all unadjusted counts. Sound good?

And then we determined that they actually were not Pavlicek points...

For those interested, I have the chance that the opps have a game as well. Shape Game? Error Opps? Error 2-2=5-4 24% 0.3% 35% 0.3% 3-2=4-4 24% 0.2% 32% 0.2% 3-3=4-3 25% 0.2% 31% 0.2% 4-3=3-3 25% 0.2% 29% 0.2% 3-2=5-3 26% 0.2% 33% 0.2% 4-2=4-3 26% 0.1% 30% 0.1% 3-3=5-2 27% 0.2% 32% 0.2% 4-3=4-2 27% 0.1% 28% 0.1% 2-2=6-3 27% 0.4% 38% 0.4% 4-4=3-2 28% 0.2% 26% 0.2% 3-2=6-2 28% 0.3% 36% 0.3% 4-2=5-2 28% 0.2% 32% 0.2% 5-2=3-3 28% 0.2% 31% 0.2% 5-3=3-2 28% 0.2% 29% 0.2% 5-2=4-2 29% 0.2% 31% 0.2% 3-1=5-4 29% 0.3% 33% 0.3% 2-2=7-2 29% * 0.7% 41% * 0.8% 2-1=6-4 30% 0.4% 38% 0.5% 2-1=5-5 30% * 0.5% 35% * 0.5% 4-1=4-4 31% 0.3% 30% 0.3% 3-1=6-3 31% 0.4% 36% 0.4% 5-4=2-2 31% 0.3% 27% 0.3% 2-1=7-3 32% * 0.7% 41% * 0.7% 4-1=5-3 32% 0.3% 32% 0.3% 3-3=6-1 32% * 0.5% 31% * 0.5% 6-2=3-2 32% 0.3% 34% 0.3% 6-3=2-2 33% 0.4% 32% 0.4% 4-3=5-1 33% 0.3% 28% 0.3% 5-1=4-3 34% 0.3% 31% 0.3% 2-2=8-1 34% * 2.2% 46% * 2.3% 3-1=7-2 34% * 0.7% 42% * 0.7% 4-4=4-1 34% 0.3% 25% 0.3% 5-3=4-1 35% 0.3% 27% 0.3% 3-2=7-1 35% * 0.7% 37% * 0.7% 4-2=6-1 35% 0.5% 32% 0.4% 4-1=6-2 35% 0.4% 35% 0.4% 5-4=3-1 37% 0.3% 25% 0.2% 2-1=8-2 37% * 1.6% 47% * 1.6% 6-1=3-3 37% * 0.5% 35% * 0.5% 6-3=3-1 38% 0.4% 29% 0.4% 5-2=5-1 38% 0.4% 30% 0.4% 7-2=2-2 38% * 0.8% 37% * 0.8% 1-1=6-5 38% * 1.2% 40% * 1.2% 5-1=5-2 39% 0.4% 33% 0.4% 3-0=5-5 39% * 1.1% 33% * 1.0% 6-1=4-2 40% 0.5% 35% 0.5% 1-1=7-4 40% * 1.7% 40% * 1.7% 6-2=4-1 41% 0.5% 32% 0.4% 3-0=6-4 41% * 0.9% 36% * 0.9% 4-0=5-4 42% * 0.6% 31% * 0.6% 6-4=2-1 43% 0.5% 28% 0.4% 4-0=6-3 43% * 0.9% 33% * 0.8% 3-0=7-3 43% * 1.4% 38% * 1.4% 7-2=3-1 43% * 0.7% 37% * 0.7% 7-1=3-2 43% * 0.7% 38% * 0.7% 5-5=2-1 43% * 0.6% 26% 0.5% 7-3=2-1 44% * 0.8% 34% * 0.7% 5-0=4-4 44% * 0.9% 31% * 0.8% 4-1=7-1 44% * 1.2% 36% * 1.1% 3-3=7-0 44% * 2.0% 29% * 1.8% 4-4=5-0 45% * 0.9% 24% * 0.8% 3-1=8-1 45% * 2.1% 44% * 2.1% 2-0=7-4 45% * 1.7% 38% * 1.7% 2-0=6-5 45% * 1.3% 36% * 1.2% 4-3=6-0 45% * 0.9% 28% * 0.8% 5-0=5-3 46% * 0.8% 32% * 0.7% 1-1=8-3 46% * 2.9% 44% * 2.9% 3-0=8-2 47% * 3.1% 46% * 3.1% 5-3=5-0 48% * 0.8% 26% * 0.7% 5-1=6-1 48% * 0.9% 35% * 0.8% 5-4=4-0 49% * 0.7% 23% * 0.5% 2-0=8-3 50% * 3.1% 42% * 3.1% 6-0=4-3 50% * 0.9% 31% * 0.8% 6-3=4-0 51% * 0.9% 26% * 0.8% 4-2=7-0 51% * 1.7% 32% * 1.6% 3-2=8-0 51% * 3.2% 36% * 3.1% 6-1=5-1 51% * 0.9% 35% * 0.8% 8-1=2-2 52% * 2.2% 45% * 2.2% 4-0=7-2 52% * 1.7% 35% * 1.6% 5-5=3-0 52% * 1.1% 22% * 0.9% 6-4=3-0 52% * 0.9% 25% * 0.8% 8-2=2-1 53% * 1.7% 42% * 1.7% 5-2=6-0 53% * 1.3% 31% * 1.2% 7-0=3-3 54% * 1.9% 38% * 1.9% 5-0=6-2 55% * 1.3% 36% * 1.2% 1-0=7-5 55% * 3.0% 41% * 3.0% 6-2=5-0 56% * 1.3% 27% * 1.1% 7-3=3-0 57% * 1.4% 29% * 1.3% 6-0=5-2 57% * 1.3% 35% * 1.2% 7-1=4-1 57% * 1.2% 35% * 1.1% 4-1=8-0 57% * 4.6% 43% * 4.6% 1-0=6-6 58% * 3.7% 44% * 3.7% 6-5=1-1 58% * 1.2% 27% * 1.1% 7-4=1-1 59% * 1.6% 30% * 1.5% 4-0=8-1 59% * 4.7% 36% * 4.6% 7-0=4-2 60% * 1.7% 34% * 1.6% 7-2=4-0 60% * 1.7% 31% * 1.6% 8-0=3-2 60% * 3.1% 43% * 3.1% 8-1=3-1 61% * 2.1% 41% * 2.1% 6-5=2-0 61% * 1.2% 25% * 1.1% 1-0=8-4 63% * 4.7% 48% * 4.9% 7-4=2-0 63% * 1.6% 27% * 1.5% 8-2=3-0 63% * 3.0% 35% * 3.0% 5-0=7-1 64% * 2.8% 42% * 2.9% 8-3=1-1 65% * 2.7% 32% * 2.6% 5-1=7-0 65% * 3.0% 33% * 3.0% 8-3=2-0 67% * 2.9% 36% * 3.0% 6-1=6-0 70% * 2.5% 35% * 2.5% 7-0=5-1 70% * 2.8% 37% * 3.0% 6-0=6-1 71% * 2.4% 38% * 2.6% 7-5=1-0 72% * 2.9% 30% * 3.0% 8-0=4-1 73% * 4.3% 41% * 4.8% 7-1=5-0 73% * 2.8% 35% * 3.0% 8-1=4-0 74% * 4.0% 44% * 4.5% 8-4=1-0 75% * 4.3% 28% * 4.4% 6-6=1-0 83% * 3.1% 24% * 3.5% There is less of a dependence on overall shape, but still a major factor if we have major cards or not. Is this a case against some assumed-fit preempts that show both majors? The opps are much less likely to have a game that we need to sacrifice against.

Welcome to these forums, Thomas.

Yes! Right here: [improving Hand Evaluation Part 1] http://tinyurl.com/25huc [improving Hand Evaluation Part 2] http://tinyurl.com/383e6 Everything is in terms of tricks, so there is no discussion of Zar, TSP, BUMRAP, or anything else. Tysen

No, it just evaluates the average value opposite all the potential hands partner could have. Here's the problem I have with your argument that my comparison doesn't take into account the potential of adjusting once a fit is found/not found. You can use the same adjusting principle with any evaluation scheme, not just Zar. No matter what system you use, you can adjust up and down the same way in order to improve it once you have more information. The point of judging the initial values to get as close as you can to begin with so that the amount you have to adjust is kept to a minimum. Sometimes the opps interfere and you can't exchange all the info you'd like. So try to make the best guess you can now instead of relying on info you might never get. Tysen

I don't really either (is that sacrilegious?) :rolleyes: Believe it or not, I'm not really interested in finding the perfect evaluator. What I am interested in is how evaluations change as you gather information from the bidding. So I need an accurate evaluator as a beginning so that I can measure changes. I've found lots of interesting stuff in my studies and I've posted some of it here and on RGB. Stuff like: How much better is Qxx in partner's suit better than Qxx in a side suit? Does Axx improve by the same amount or different? How much should we adjust the value of Kxx if RHO opens the suit? If LHO opens it? How the relative weight of high cards to shape changes. If partner is balanced, high cards gain more weight and shape loses importance. If partner is unbalanced, high cards lose weight and shape gains. The same is true if the opps are balanced/unbalanced, and to a different extent. Tysen

Okay, I can see where you got this, but there is no need for 0 points to equal 0 tricks. Using 2.5 points per trick would also predict that you would need only 30 points for slam and 32.5 for a grand. That's obviously off. The real value should be slightly bigger than 3 I'd say. And the scale for TSP (which uses the length points too) is 5 points per trick. Also, Ben, you've recreated a table I used way back on page 5 of this super-long thread. Same metrics using tricks.

I was thinking about this yesterday too. On my list.

The database is 1 million random deals that Matt Ginsberg compiled using GIB. The database is available on his website. I don't know how long it took him to generate them all.

To copy the metric being developed in the other thread on DD evaluations (which is starting to generate some interesting discussion), let's see how it applies to shapes as a whole. Given the shape of our hand, what is the chance that we have a game? Shape Game? 531 Zar 4-3-3-3 25% 4-4-3-2 26% 5-3-3-2 27% 5-4-2-2 28% 6-3-2-2 30% 4-4-4-1 32% <-- 5-4-3-1 33% <-- 7-2-2-2 34% <-- <-- 6-3-3-1 34% <-- <-- 6-4-2-1 37% 5-5-2-1 38% <-- 7-3-2-1 38% 5-4-4-0 45% <-- 5-5-3-0 46% 6-4-3-0 47% 6-5-1-1 49% 7-4-1-1 50% 6-5-2-0 54% 7-4-2-0 55% The first set of arrows shows hands that a 5-3-1 system counts as "equivalent." The second set of arrows shows some hands that Zar counts as equivalent. Which set looks more tightly clustered to you?

Okay, then is this 2 or 4 superfit points if there is no opener? xxxxx x xxx xxxx xxxxx xx xxx xxx

Is that not enough? Maybe the question would be easier to answer if you asked "Why do we pass?" rather than "Why do we open?"? Tysen

Okay here is a more complete list: Shape Game? Error 2-2=5-4 24% 0.3% 3-2=4-4 24% 0.2% 3-3=4-3 25% 0.2% 4-3=3-3 25% 0.2% 3-2=5-3 26% 0.2% 4-2=4-3 26% 0.1% 3-3=5-2 27% 0.2% 4-3=4-2 27% 0.1% 2-2=6-3 27% 0.4% 4-4=3-2 28% 0.2% 3-2=6-2 28% 0.3% 4-2=5-2 28% 0.2% 5-2=3-3 28% 0.2% 5-3=3-2 28% 0.2% 5-2=4-2 29% 0.2% 3-1=5-4 29% 0.3% 2-2=7-2 29% * 0.7% 2-1=6-4 30% 0.4% 2-1=5-5 30% * 0.5% 4-1=4-4 31% 0.3% 3-1=6-3 31% 0.4% 5-4=2-2 31% 0.3% 2-1=7-3 32% * 0.7% 4-1=5-3 32% 0.3% 3-3=6-1 32% * 0.5% 6-2=3-2 32% 0.3% 6-3=2-2 33% 0.4% 4-3=5-1 33% 0.3% 5-1=4-3 34% 0.3% 2-2=8-1 34% * 2.2% 3-1=7-2 34% * 0.7% 4-4=4-1 34% 0.3% 5-3=4-1 35% 0.3% 3-2=7-1 35% * 0.7% 4-2=6-1 35% 0.5% 4-1=6-2 35% 0.4% 5-4=3-1 37% 0.3% 2-1=8-2 37% * 1.6% 6-1=3-3 37% * 0.5% 6-3=3-1 38% 0.4% 5-2=5-1 38% 0.4% 7-2=2-2 38% * 0.8% 1-1=6-5 38% * 1.2% 5-1=5-2 39% 0.4% 3-0=5-5 39% * 1.1% 6-1=4-2 40% 0.5% 1-1=7-4 40% * 1.7% 6-2=4-1 41% 0.5% 3-0=6-4 41% * 0.9% 4-0=5-4 42% * 0.6% 6-4=2-1 43% 0.5% 4-0=6-3 43% * 0.9% 3-0=7-3 43% * 1.4% 7-2=3-1 43% * 0.7% 7-1=3-2 43% * 0.7% 5-5=2-1 43% * 0.6% 7-3=2-1 44% * 0.8% 5-0=4-4 44% * 0.9% 4-1=7-1 44% * 1.2% 3-3=7-0 44% * 2.0% 4-4=5-0 45% * 0.9% 3-1=8-1 45% * 2.1% 2-0=7-4 45% * 1.7% 2-0=6-5 45% * 1.3% 4-3=6-0 45% * 0.9% 5-0=5-3 46% * 0.8% 1-1=8-3 46% * 2.9% 3-0=8-2 47% * 3.1% 5-3=5-0 48% * 0.8% 5-1=6-1 48% * 0.9% 5-4=4-0 49% * 0.7% 2-0=8-3 50% * 3.1% 6-0=4-3 50% * 0.9% 6-3=4-0 51% * 0.9% 4-2=7-0 51% * 1.7% 3-2=8-0 51% * 3.2% 6-1=5-1 51% * 0.9% 8-1=2-2 52% * 2.2% 4-0=7-2 52% * 1.7% 5-5=3-0 52% * 1.1% 6-4=3-0 52% * 0.9% 8-2=2-1 53% * 1.7% 5-2=6-0 53% * 1.3% 7-0=3-3 54% * 1.9% 5-0=6-2 55% * 1.3% 1-0=7-5 55% * 3.0% 6-2=5-0 56% * 1.3% 7-3=3-0 57% * 1.4% 6-0=5-2 57% * 1.3% 7-1=4-1 57% * 1.2% 4-1=8-0 57% * 4.6% 1-0=6-6 58% * 3.7% 6-5=1-1 58% * 1.2% 7-4=1-1 59% * 1.6% 4-0=8-1 59% * 4.7% 7-0=4-2 60% * 1.7% 7-2=4-0 60% * 1.7% 8-0=3-2 60% * 3.1% 8-1=3-1 61% * 2.1% 6-5=2-0 61% * 1.2% 1-0=8-4 63% * 4.7% 7-4=2-0 63% * 1.6% 8-2=3-0 63% * 3.0% 5-0=7-1 64% * 2.8% 8-3=1-1 65% * 2.7% 5-1=7-0 65% * 3.0% 8-3=2-0 67% * 2.9% 6-1=6-0 70% * 2.5% 7-0=5-1 70% * 2.8% 6-0=6-1 71% * 2.4% 7-5=1-0 72% * 2.9% 8-0=4-1 73% * 4.3% 7-1=5-0 73% * 2.8% 8-1=4-0 74% * 4.0% 8-4=1-0 75% * 4.3% 6-6=1-0 83% * 3.1% If there is a "*" that means the error is more than 0.5% and so the value should be looked at with a grain of salt. Look at some of the ones that were excluded before. 2-2=7-2 stands out like a sore thumb. It rarely produces a game and is worse than all 4441 shapes.

Yes. And while HCP & Shape are slightly correlated, when you add up every single HCP possibility, you get everything.

5-0=4-4 was cut off from the list. It is 44%. I'll edit the original post to include it. I cut off from the table everything that happened less than 5000 times in my database. So 5-0=4-4 just happened to be below that while the other 5440 patterns were above. At a count of 5000, the error is on the order of 0.5%, so I didn't want to include rarer patterns. Let me know if any other patterns are missing. Tysen

Okay, here we go: Shape Game? 2-2=5-4 24% 3-2=4-4 24% 3-3=4-3 25% 4-3=3-3 25% 3-2=5-3 26% 4-2=4-3 26% 3-3=5-2 27% 4-3=4-2 27% 2-2=6-3 27% 4-4=3-2 28% 3-2=6-2 28% 4-2=5-2 28% 5-2=3-3 28% 5-3=3-2 28% 5-2=4-2 29% 3-1=5-4 29% 2-1=6-4 30% 2-1=5-5 30% 4-1=4-4 31% 3-1=6-3 31% 5-4=2-2 31% 4-1=5-3 32% 3-3=6-1 32% 6-2=3-2 32% 6-3=2-2 33% 4-3=5-1 33% 5-1=4-3 34% 4-4=4-1 34% 5-3=4-1 35% 4-2=6-1 35% 4-1=6-2 35% 5-4=3-1 37% 6-1=3-3 37% 6-3=3-1 38% 5-2=5-1 38% 5-1=5-2 39% 6-1=4-2 40% 6-2=4-1 41% 4-0=5-4 42% 6-4=2-1 43% 5-5=2-1 43% 5-0=4-4 44% 5-4=4-0 49% Edit: A more complete list is now available on the 2nd page of this thread. Because of what we're looking for, S&H are equivelent, as are D&C. So I'm going to use the notation A-B=C-D to be A&B in the majors and C&D in the minors (either way for both). So maybe 2245 is the worst distribution in bridge and not 3334! ;) Okay, what can we make of this? It's a very loose scale, but 1 point of "normal" distribution corresponds to about 3.8% game increase. On Zar's scale, it's about 1 point per 2.3%. So that means that the difference between 4-4=4-1 and 4-1=4-4 is about 1.0 points or 1.6 Zar. To address awm's original guesses, the difference between 6-3=3-1 and 6-3=2-2 is about the same as the difference between 3-3=6-1 and 3-2=6-2 (about 1.2 points or 2.0 Zar in both cases). And the difference between 4-4=3-2 and 4-2=4-3 is only 0.5 points or 0.8 Zar. There are some other interesting ramifications here, but I don't want to hog the stage. Anyone want to comment? Tysen

You guys are all pretty much right. I just want to state that: 1. Zar, BUMRAP, and TSP all yield very similar results. That is because the main benefit from these methods comes from shifting the HCP values from a 4-3-2-1 ratio to a 3-2-1-0.5 ratio. The distribution is fairly minor compared to this. 2. Zar is a perfectly good system and much better than regular HCP. My single and only complaint about Zar has been that it's slightly more complicated to calculate than BUM+531 and it's not any more accurate. Plus it uses a completely different "scale" which some people don't want to use and makes it more difficult to explain to opponents. 3. The only reason I invented TSP was that I said to myself, "what is the most accurate point count method I can make using reasonably sized whole numbers?" And TSP came out of that. It's not that much better than Zar or BUM, but it's the best I could do. If you want a simple method that has the same scale, just use BUMRAP. After all, if you have this hand: ATxxx Axx ATx xx It's much easier to explain that you've decided to upgrade this hand to 15 points rather than explain that you've got 29 Zar. Tysen

I don't have the % of time it makes game, but I do have the average number of tricks that each pattern makes. It doesn't differentiate between majors and minors. 4-3-3-3 7.80 4-4-3-2 8.09 5-3-3-2 8.14 5-4-2-2 8.41 6-3-2-2 8.51 4-4-4-1 8.62 5-4-3-1 8.69 6-3-3-1 8.78 7-2-2-2 8.91 6-4-2-1 9.02 5-5-2-1 9.03 7-3-2-1 9.14 5-4-4-0 9.38 5-5-3-0 9.51 6-4-3-0 9.51 8-2-2-1 9.57 6-5-1-1 9.61 7-3-3-0 9.65 7-4-1-1 9.67 8-3-1-1 9.83 6-5-2-0 9.88 7-4-2-0 9.89 But the suggestion you make is an interesting one. I think I could whip it up and take a look at the %game statistic. That would differentiate between majors and minors and see how much a 1444 hand is compared to a 4441 hand. Good idea.