The tabulated simulation results.
Total games played=10000000, total rounds=259722851,
took 7204 seconds. Games that never needed 15 cards
on the table: 3124823 (31.2 percent of all games).
Games where no cards remain on the table at the end:
118583 (1.2 percent of all games).
Selection mode="Random among 'most similar' Sets" (3),
seed=112881665
In deck | Set | NoSet | Set:NoSet for 12 | Avg # of Sets
--------+--------+-------+------------------+-------------
69 |9677946 |322054 | 30.1:1 | 2.78
66 |9470146 |526289 | 18.0:1 | 2.53
63 |9373021 |621021 | 15.1:1 | 2.43
60 |9335349 |657731 | 14.2:1 | 2.39
57 |9321017 |671573 | 13.9:1 | 2.38
54 |9317330 |675273 | 13.8:1 | 2.38
51 |9314262 |678493 | 13.7:1 | 2.38
48 |9311388 |681061 | 13.7:1 | 2.37
45 |9309101 |683320 | 13.6:1 | 2.37
42 |9306497 |685766 | 13.6:1 | 2.37
39 |9304125 |688201 | 13.5:1 | 2.36
36 |9300767 |691513 | 13.4:1 | 2.36
33 |9297844 |694237 | 13.4:1 | 2.36
30 |9294812 |697142 | 13.3:1 | 2.36
27 |9293372 |698631 | 13.3:1 | 2.35
24 |9290828 |701190 | 13.3:1 | 2.35
21 |9289767 |702063 | 13.2:1 | 2.35
18 |9288336 |703590 | 13.2:1 | 2.35
15 |9287394 |704505 | 13.2:1 | 2.35
12 |9287464 |704377 | 13.2:1 | 2.35
9 |9287512 |704198 | 13.2:1 | 2.35
6 |9286622 |705358 | 13.2:1 | 2.35
3 |9287553 |704197 | 13.2:1 | 2.35
0 |9250663 |741608 | 12.5:1 | 2.35
In deck | Set | NoSet | Set:NoSet for 15 | Avg # of Sets
--------+--------+-------+------------------+-------------
69 | 0 | 0 | oo:1 | -
66 | 318489 | 3565 | 89.3:1 | 3.33
63 | 523896 | 5958 | 87.9:1 | 3.35
60 | 620059 | 6920 | 89.6:1 | 3.35
57 | 657241 | 7408 | 88.7:1 | 3.36
54 | 671586 | 7395 | 90.8:1 | 3.36
51 | 675425 | 7245 | 93.2:1 | 3.36
48 | 678187 | 7551 | 89.8:1 | 3.36
45 | 681033 | 7578 | 89.9:1 | 3.36
42 | 683162 | 7736 | 88.3:1 | 3.35
39 | 685829 | 7673 | 89.4:1 | 3.35
36 | 688155 | 7720 | 89.1:1 | 3.35
33 | 691314 | 7919 | 87.3:1 | 3.34
30 | 694110 | 8046 | 86.3:1 | 3.34
27 | 697191 | 7997 | 87.2:1 | 3.34
24 | 698646 | 7982 | 87.5:1 | 3.33
21 | 701002 | 8170 | 85.8:1 | 3.34
18 | 702159 | 8073 | 87.0:1 | 3.33
15 | 703563 | 8101 | 86.8:1 | 3.33
12 | 704447 | 8159 | 86.3:1 | 3.33
9 | 704246 | 8290 | 85.0:1 | 3.33
6 | 704468 | 8020 | 87.8:1 | 3.33
3 | 705128 | 8250 | 85.5:1 | 3.33
0 | 704718 | 7726 | 91.2:1 | 3.33
In deck | Set | NoSet | Set:NoSet for 18 | Avg # of Sets
--------+--------+-------+------------------+-------------
69 | 0 | 0 | oo:1 | -
66 | 0 | 0 | oo:1 | -
63 | 3565 | 0 | oo:1 | 5.31
60 | 5958 | 0 | oo:1 | 5.36
57 | 6918 | 2 | 3459.0:1 | 5.35
54 | 7408 | 2 | 3704.0:1 | 5.37
51 | 7397 | 0 | oo:1 | 5.36
48 | 7245 | 0 | oo:1 | 5.39
45 | 7550 | 1 | 7550.0:1 | 5.35
42 | 7578 | 1 | 7578.0:1 | 5.35
39 | 7736 | 1 | 7736.0:1 | 5.33
36 | 7674 | 0 | oo:1 | 5.34
33 | 7720 | 0 | oo:1 | 5.32
30 | 7919 | 0 | oo:1 | 5.33
27 | 8046 | 0 | oo:1 | 5.29
24 | 7997 | 0 | oo:1 | 5.32
21 | 7982 | 0 | oo:1 | 5.34
18 | 8169 | 1 | 8169.0:1 | 5.31
15 | 8074 | 0 | oo:1 | 5.32
12 | 8101 | 0 | oo:1 | 5.32
9 | 8159 | 0 | oo:1 | 5.32
6 | 8290 | 0 | oo:1 | 5.32
3 | 8020 | 0 | oo:1 | 5.34
0 | 8247 | 3 | 2749.0:1 | 5.32
In deck | Set | NoSet | Set:NoSet for 21 | Avg # of Sets
--------+--------+-------+------------------+-------------
69 | 0 | 0 | oo:1 | -
66 | 0 | 0 | oo:1 | -
63 | 0 | 0 | oo:1 | -
60 | 0 | 0 | oo:1 | -
57 | 0 | 0 | oo:1 | -
54 | 2 | 0 | oo:1 | 7.50
51 | 2 | 0 | oo:1 | 7.50
48 | 0 | 0 | oo:1 | -
45 | 0 | 0 | oo:1 | -
42 | 1 | 0 | oo:1 | 10.00
39 | 1 | 0 | oo:1 | 8.00
36 | 1 | 0 | oo:1 | 9.00
33 | 0 | 0 | oo:1 | -
30 | 0 | 0 | oo:1 | -
27 | 0 | 0 | oo:1 | -
24 | 0 | 0 | oo:1 | -
21 | 0 | 0 | oo:1 | -
18 | 0 | 0 | oo:1 | -
15 | 1 | 0 | oo:1 | 7.00
12 | 0 | 0 | oo:1 | -
9 | 0 | 0 | oo:1 | -
6 | 0 | 0 | oo:1 | -
3 | 0 | 0 | oo:1 | -
0 | 0 | 0 | oo:1 | -
Henrik Warne's blog 
Hi,
can you calculate a probability of 0,6,9,12,15,18 cards left in the end of the game? I’ve found: 1% for 0 (clean up) in your estimates, 18: 1% (93:1 for no set in 15 cards with 0 cards in deck), 15: 7% (13.5:1 for no set in 12 cards with 0 cards in deck). But I’m curios about 9,12,15 as well.
For 1,000,000 games played:
21 left on table: 0 (0,0 percent of all games)
18 left on table: 0 (0,0 percent of all games)
15 left on table: 798 (0,1 percent of all games)
12 left on table: 73651 (7,4 percent of all games)
9 left on table: 447233 (44,7 percent of all games)
6 left on table: 466554 (46,7 percent of all games)
3 left on table: 0 (0,0 percent of all games)
0 left on table: 11764 (1,2 percent of all games)
If there are only 3 cards left, those 3 cards
also form a set, therefore that value is zero
(but you already knew that :-)).
Thanks, that pretty good correlates with real data for ~10k games on my site:
18: 0%
15: 0.0747%
12: 7.3936%
9: 43.4902%
6: 48.2325%
0: 1.1949%
I thought that dataset is too small for any objective result, but seems as it’s ok.