I also found that re-ordering the tables from most probable to least probable, as below, reveals some obvious patterns, many of which I hadn't noticed before (perhaps this shows how unobservant I am). I omitted 2 card suit divisions for 2 reasons: they break the pattern, and you should be able to calculate them on the fly.
1. x-y divisions are grouped by the difference between x&y:
1 off (e.g. 3-2, 4-3) is around 2/3.
2 off is around 50-50.
Even is 1/3+.
Off by 3 around 30%, etc.
(this is increasing except for exactly even)
If you forget 4-3, it's going to be closer to 3-2 than any other number you remember. Also, these groups don't overlap (excluding 2 card suits or 11+ card suits, and 5-5 is barely less likely than 6-3)
2. For the close divisions (not more than 2 off), the more cards there are the more likely you are to be off by more than 2, so the groups go from short suits to long suits (e.g. 4-2 is more likely than 5-3). After that, longer suits are more likely than shorter suits (e.g. 5-2 is more likely than 4-1).
3. As suits get longer, the odds become more similar. For example, 6-3 is virtually the same odds as 5-2. In contrast, 2-1 is 10% more likely than 3-2. If you remember 2 or 3 odds for same difference combinations, you can generally extrapolate the others. Generally, the differences cut about in half each time. For example, 3-2 / 4-3 are 68 / 62, so 5-4 is probably around 62 - 1/2 (68-62) = 59% (in fact it is).
Some more stuff below the table...
| 2-1 | 78% |
| 3-2 | 68% |
| 4-3 | 62% |
| 3-1 | 50% |
| 4-2 | 48% |
| 5-3 | 47% |
| 2-2 | 41% |
| 3-3 | 36% |
| 4-4 | 33% |
| 5-2 | 31% |
| 4-1 | 28% |
| 3-0 | 22% |
| 6-2 | 17% |
| 5-1 | 15% |
| 4-0 | 10% |
| 7-2 | 9% |
| 6-1 | 7% |
| 5-0 | 4% |
4. Void probabilities are relatively easy to derive from one another just by using open spaces:
1-0 is 100%,
2-0 is 12/25 of that, 48%,
3-0 is 11/24 of that 22%,
4-0 is 10/23 of that, a bit under 10% (9.6%),
5-0 is 9/22 of that, 3.9%,
6-0 is 8/21 of that, about 1.5%.
5. The singletons are harder but doable in a pinch (adjust for open spaces then number of possible singletons):
2-1 is 78%
3-1 is 11/23 * 4/3 of that, about 50%
4-1 is 10/22 * 5/4 of that, about 28%
5-1 is 9/21 * 6/5 of that, about 14%
6-1 is 8/20 * 7/6 of that, about 7%
So...
If you can figure out short suits and maybe some voids on the fly and remember these:
3-2 68%
4-2 48%
3-3 35.5%
4-1 28%
4-0 10%
You can probably work out the rest. 4-3? Well, 2-1 is 78 and 3-2 68 so subtract about half that difference from 68 to get 63 (actually 62). 5-3? All the off by 2s are right around 48%, should be a bit less (in fact 47). 5-2? 3-0 is 22, 4-1 is 28, so add half the difference to 28 and get 31 (in fact 30.5). 4-4? 2-2 is 41, 3-3 is 35.5, subtract half the difference and get almost 33 (in fact 32.7).
6-2 is maybe a bit harder. We memorized 4-0 as 10%. 5-1 is about half of 4-1 (9/21 * 6/5 to be precise), call it 14% (we can check this also by noting that 3-3 and 4-2 add to 84, so it should be a touch less than 16%). 6-2 is probably around 16% (in fact, 17.1. 4-0 is a bit less than 10% and 5-1 is 14.5, so my extrapolation suffered from some rounding).
You could also get there by noting that 5-3 and 4-4 add to 80%, so 6-2 ought to be most of the rest. Delving a bit further, you could consider there are 28 6-2 layouts vs 8 7-1. The latter are about half as likely, also (the last card has 12 open spaces vs 7), so call it 7:1, so the 7-1s are about 1/8 of 20% and the 6-2s are the rest (8-0 never happens, right?): 17.5%
The harder of these are not that practical, but trying to estimate these numbers and occasionally checking your results will result in better memorization.
Nice information!
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