Study 8J53   Main

Running Suits

by Richard Pavlicek

Joe Foster of Framingham, Massachusetts, offered this poser:

Below are four possible suit holdings in one hand in a notrump contract. What are the respective probabilities that the suits will run (no losers) if you know nothing about the other three hands. Treat this as single-dummy, played for maximum tricks, with sufficient side entries (if necessary) in both declarer’s and dummy’s hand.

A-K-Q-6-5-4-3-2

A-K-Q-5-4-3-2

A-K-Q-4-3-2

A-K-Q-3-2

The impulsive answer in each case is “Who really cares?” but determining exact theoretical chances has a curious appeal. Obviously an eight-card suit would be quite high, probably in the mid 90s percent, then progressively lower, with a five-card suit maybe in the mid 60s. Alas, ballpark assessments won’t cut it here, so let’s do some number crunching.

As a computational aid I used my Three-Hand Suit Break Calculator to obtain ratios for each possible suit division.

A-K-Q-6-5-4-3-2

South holding an eight-card suit is easy to calculate. The only suit divisions with partial success are 4-1-0 and 0-1-4, which succeed only when North’s singleton is the jack, hence 1/5 of the time. Rather than introduce fractions, I multiplied each ratio by an appropriate factor (5 = complete success) to obtain the following table:

W-N-E	North Success	Factor	Ratio	Product
2-2-1	all	5	6084	30420
2-1-2	all	5	6084	30420
1-2-2	all	5	6084	30420
3-1-1	all	5	3718	18590
1-3-1	all	5	3718	18590
1-1-3	all	5	3718	18590
3-2-0	all	5	1716	8580
3-0-2	all	5	1716	8580
2-3-0	all	5	1716	8580
2-0-3	all	5	1716	8580
0-3-2	all	5	1716	8580
0-2-3	all	5	1716	8580
4-1-0	J	1	715	715
4-0-1	none	0	715	0
1-4-0	all	5	715	3575
1-0-4	none	0	715	0
0-4-1	all	5	715	3575
0-1-4	J	1	715	715
5-0-0	none	0	99	0
0-5-0	all	5	99	495
0-0-5	none	0	99	0
Successful total				207585
Possible total		5	44289	221445
Successful percent		93.7411095305832148

Study 8J53   Main

Top   Running Suits

A-K-Q-5-4-3-2

South holding a seven-card suit is almost as easy. Four divisions allow partial success: 4-1-1 or 1-1-4 with North having a singleton jack, which is 1/6 of the time; and 4-2-0 or 0-2-4 with North having J-x, which is 1/3 of the time. Again to eliminate fractions, I used the factor 6 to represent complete success, producing this table:

W-N-E	North Success	Factor	Ratio	Product
2-2-2	all	6	36504	219024
3-2-1	all	6	22308	133848
3-1-2	all	6	22308	133848
2-3-1	all	6	22308	133848
2-1-3	all	6	22308	133848
1-3-2	all	6	22308	133848
1-2-3	all	6	22308	133848
4-1-1	J	1	9295	9295
1-4-1	all	6	9295	55770
1-1-4	J	1	9295	9295
3-3-0	all	6	6292	37752
3-0-3	all	6	6292	37752
0-3-3	all	6	6292	37752
4-2-0	J-x	2	4290	8580
4-0-2	none	0	4290	0
2-4-0	all	6	4290	25740
2-0-4	none	0	4290	0
0-4-2	all	6	4290	25740
0-2-4	J-x	2	4290	8580
5-1-0	none	0	1287	0
5-0-1	none	0	1287	0
1-5-0	all	6	1287	7722
1-0-5	none	0	1287	0
0-5-1	all	6	1287	7722
0-1-5	none	0	1287	0
6-0-0	none	0	132	0
0-6-0	all	6	132	792
0-0-6	none	0	132	0
Successful total				1294604
Possible total		6	250971	1505826
Successful percent		85.9730141463887594

Did you hear about the physicist who hailed from Newton, Mass?
He accelerated his bridge studies and became a force to be reckoned with.

Study 8J53   Main

Top   Running Suits

A-K-Q-4-3-2

South holding a six-card suit adds some complications. The distribution 4-2-1 or 1-2-4 succeeds when North has the jack or 10-x if the jack is singleton, comprising 1/3 of the combinations. While 4-3-0 is straightforward whenever North has the jack (3/7 of the time), 0-3-4 does better, also succeeding with 10-9-x (finessing after West shows out) which amounts to 19 of 35 combinations. Another oddball is 5-2-0 or 0-2-5, succeeding only with J-10 doubleton, 1/21 of the time. The lowest common denominator to eliminate fractions is now 105, producing:

W-N-E	North Success	Factor	Ratio	Product
3-2-2	all	105	12168	1277640
2-3-2	all	105	12168	1277640
2-2-3	all	105	12168	1277640
3-3-1	all	105	7436	780780
3-1-3	all	105	7436	780780
1-3-3	all	105	7436	780780
4-2-1	J-x, 10-x + J East	35	5070	177450
4-1-2	J	15	5070	76050
2-4-1	all	105	5070	532350
2-1-4	J	15	5070	76050
1-4-2	all	105	5070	532350
1-2-4	J-x, 10-x + J West	35	5070	177450
5-1-1	none	0	1521	0
1-5-1	all	105	1521	159705
1-1-5	none	0	1521	0
4-3-0	J-x-x	45	1430	64350
4-0-3	none	0	1430	0
3-4-0	all	105	1430	150150
3-0-4	none	0	1430	0
0-4-3	all	105	1430	150150
0-3-4	J-x-x, 10-9-x	57	1430	81510
5-2-0	J-10	5	702	3510
5-0-2	none	0	702	0
2-5-0	all	105	702	73710
2-0-5	none	0	702	0
0-5-2	all	105	702	73710
0-2-5	J-10	5	702	3510
6-1-0	none	0	156	0
6-0-1	none	0	156	0
1-6-0	all	105	156	16380
1-0-6	none	0	156	0
0-6-1	all	105	156	16380
0-1-6	none	0	156	0
7-0-0	none	0	12	0
0-7-0	all	105	12	1260
0-0-7	none	0	12	0
Successful total				8541285
Possible total		105	107559	11293695
Successful percent		75.6287911086672697

Study 8J53   Main

Top   Running Suits

A-K-Q-3-2

South with just five cards brings in some new twists. The distribution 3-2-3 might seem a complete success, but no; the proper play with 10-9 doubleton is to run the 10, failing when West has J-x-x, so you succeed only 55/56 of the time. Similarly, 4-2-2 or 2-2-4 not only succeeds when North has the jack, but with 10-9 doubleton if East has the jack. Another curiosity is 0-4-4, succeeding when North has J-x-x-x, 10-9-x-x, 10-8-7-x or 9-8-7-x, the last three by finessing after West shows out. The lowest common denominator keeps growing and is now 840, producing this table:

W-N-E	North Success	Factor	Ratio	Product
3-3-2	all	840	44616	37477440
3-2-3	all but 10-9 + J West	825	44616	36808200
2-3-3	all	840	44616	37477440
4-2-2	J-x, 10-9 + J East	220	30420	6692400
2-4-2	all	840	30420	25552800
2-2-4	J-x, 10-9 + J East	230	30420	6996600
4-3-1	J-x-x, 10-x-x + J East	360	18590	6692400
4-1-3	J	105	18590	1951950
3-4-1	all	840	18590	15615600
3-1-4	J	105	18590	1951950
1-4-3	all	840	18590	15615600
1-3-4	J-x-x, 10-x-x + J West	360	18590	6692400
5-2-1	J-10, J-9 + 10 East	35	9126	319410
5-1-2	none	0	9126	0
2-5-1	all	840	9126	7665840
2-1-5	none	0	9126	0
1-5-2	all	840	9126	7665840
1-2-5	J-10, J-9 + 10 West	35	9126	319410
4-4-0	J-x-x-x	420	3575	1501500
4-0-4	none	0	3575	0
0-4-4	J-x-x-x, 10-9-x-x, 10-8-7-x, 9-8-7-x	612	3575	2187900
5-3-0	J-10-x, J-9-x	165	2574	424710
5-0-3	none	0	2574	0
3-5-0	all	840	2574	2162160
3-0-5	none	0	2574	0
0-5-3	all	840	2574	2162160
0-3-5	J-10-x	90	2574	231660
6-1-1	none	0	2028	0
1-6-1	all	840	2028	1703520
1-1-6	none	0	2028	0
6-2-0	J-10	30	936	28080
6-0-2	none	0	936	0
2-6-0	all	840	936	786240
2-0-6	none	0	936	0
0-6-2	all	840	936	786240
0-2-6	J-10	30	936	28080
7-1-0	none	0	156	0
7-0-1	none	0	156	0
1-7-0	all	840	156	131040
1-0-7	none	0	156	0
0-7-1	all	840	156	131040
0-1-7	none	0	156	0
8-0-0	none	0	9	0
0-8-0	all	840	9	7560
0-0-8	none	0	9	0
Successful total				227767170
Possible total		840	430236	361398240
Successful percent		63.0238736082389333

And in conclusion…

So the answers are, respectively, mid 90s, mid 80s, mid 70s and mid 60s, which is about what anyone might guess. Knowing the exact numbers, however, will improve your game immensely. Now you are a complete bridge player — or a complete nut case. [pick one]

Did I say exact? Well, not quite in theory, because each suit has cases that may require an outside entry (e.g., A-K-Q-4-3-2 opposite a stiff jack). Even though the conditions assume this to be available, it would have some kind of skewing effect on random distributions. The impact would be negligible for sure, like a grain of sand to Miami Beach, but it would have to be there.

Thanks to Joe Foster, not only for submitting this problem but for his own work in determining solutions. His keen eye caught an omission in my original table for A-K-Q-3-2, now fixed. Between us, I think we got it right.

Study 8J53   Main

Top   Running Suits

© 2019 Richard Pavlicek