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	30,420
2-1-2	all	5	6084	30,420
1-2-2	all	5	6084	30,420
3-1-1	all	5	3718	18,590
1-3-1	all	5	3718	18,590
1-1-3	all	5	3718	18,590
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				207,585
Possible total		5	44,289	221,445
Successful percent		93.7411095305832148

Study 8J53 Main

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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	36,504	219,024
3-2-1	all	6	22,308	133,848
3-1-2	all	6	22,308	133,848
2-3-1	all	6	22,308	133,848
2-1-3	all	6	22,308	133,848
1-3-2	all	6	22,308	133,848
1-2-3	all	6	22,308	133,848
4-1-1	J	1	9295	9295
1-4-1	all	6	9295	55,770
1-1-4	J	1	9295	9295
3-3-0	all	6	6292	37,752
3-0-3	all	6	6292	37,752
0-3-3	all	6	6292	37,752
4-2-0	J-x	2	4290	8580
4-0-2	none	0	4290	0
2-4-0	all	6	4290	25,740
2-0-4	none	0	4290	0
0-4-2	all	6	4290	25,740
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				1,294,604
Possible total		6	250,971	1,505,826
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	12,168	1,277,640
2-3-2	all	105	12,168	1,277,640
2-2-3	all	105	12,168	1,277,640
3-3-1	all	105	7436	780,780
3-1-3	all	105	7436	780,780
1-3-3	all	105	7436	780,780
4-2-1	J-x, 10-x + J East	35	5070	177,450
4-1-2	J	15	5070	76,050
2-4-1	all	105	5070	532,350
2-1-4	J	15	5070	76,050
1-4-2	all	105	5070	532,350
1-2-4	J-x, 10-x + J West	35	5070	177,450
5-1-1	none	0	1521	0
1-5-1	all	105	1521	159,705
1-1-5	none	0	1521	0
4-3-0	J-x-x	45	1430	64,350
4-0-3	none	0	1430	0
3-4-0	all	105	1430	150,150
3-0-4	none	0	1430	0
0-4-3	all	105	1430	150,150
0-3-4	J-x-x, 10-9-x	57	1430	81,510
5-2-0	J-10	5	702	3510
5-0-2	none	0	702	0
2-5-0	all	105	702	73,710
2-0-5	none	0	702	0
0-5-2	all	105	702	73,710
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	16,380
1-0-6	none	0	156	0
0-6-1	all	105	156	16,380
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				8,541,285
Possible total		105	107,559	11,293,695
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	44,616	37,477,440
3-2-3	all but 10-9 + J West	825	44,616	36,808,200
2-3-3	all	840	44,616	37,477,440
4-2-2	J-x, 10-9 + J East	220	30,420	6,692,400
2-4-2	all	840	30,420	25,552,800
2-2-4	J-x, 10-9 + J East	230	30,420	6,996,600
4-3-1	J-x-x, 10-x-x + J East	360	18,590	6,692,400
4-1-3	J	105	18,590	1,951,950
3-4-1	all	840	18,590	15,615,600
3-1-4	J	105	18,590	1,951,950
1-4-3	all	840	18,590	15,615,600
1-3-4	J-x-x, 10-x-x + J West	360	18,590	6,692,400
5-2-1	J-10, J-9 + 10 East	35	9126	319,410
5-1-2	none	0	9126	0
2-5-1	all	840	9126	7,665,840
2-1-5	none	0	9126	0
1-5-2	all	840	9126	7,665,840
1-2-5	J-10, J-9 + 10 West	35	9126	319,410
4-4-0	J-x-x-x	420	3575	1,501,500
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	2,187,900
5-3-0	J-10-x, J-9-x	165	2574	424,710
5-0-3	none	0	2574	0
3-5-0	all	840	2574	2,162,160
3-0-5	none	0	2574	0
0-5-3	all	840	2574	2,162,160
0-3-5	J-10-x	90	2574	231,660
6-1-1	none	0	2028	0
1-6-1	all	840	2028	1,703,520
1-1-6	none	0	2028	0
6-2-0	J-10	30	936	28,080
6-0-2	none	0	936	0
2-6-0	all	840	936	786,240
2-0-6	none	0	936	0
0-6-2	all	840	936	786,240
0-2-6	J-10	30	936	28,080
7-1-0	none	0	156	0
7-0-1	none	0	156	0
1-7-0	all	840	156	131,040
1-0-7	none	0	156	0
0-7-1	all	840	156	131,040
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				227,767,170
Possible total		840	430,236	361,398,240
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