Main   Study 8J53 by Richard Pavlicek  

Running Suits

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-6-5-4-3-2A-K-Q-5-4-3-2A-K-Q-4-3-2A-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-ENorth SuccessFactorRatioProduct
2-2-1all5608430420
2-1-2all5608430420
1-2-2all5608430420
3-1-1all5371818590
1-3-1all5371818590
1-1-3all5371818590
3-2-0all517168580
3-0-2all517168580
2-3-0all517168580
2-0-3all517168580
0-3-2all517168580
0-2-3all517168580
4-1-0J1715715
4-0-1none07150
1-4-0all57153575
1-0-4none07150
0-4-1all57153575
0-1-4J1715715
5-0-0none0990
0-5-0all599495
0-0-5none0990
Successful total207585
Possible total544289221445
Successful percent93.7411095305832148

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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-ENorth SuccessRatioFactorProduct
2-2-2all636504219024
3-2-1all622308133848
3-1-2all622308133848
2-3-1all622308133848
2-1-3all622308133848
1-3-2all622308133848
1-2-3all622308133848
4-1-1J192959295
1-4-1all6929555770
1-1-4J192959295
3-3-0all6629237752
3-0-3all6629237752
0-3-3all6629237752
4-2-0J-x242908580
4-0-2none042900
2-4-0all6429025740
2-0-4none042900
0-4-2all6429025740
0-2-4J-x242908580
5-1-0none012870
5-0-1none012870
1-5-0all612877722
1-0-5none012870
0-5-1all612877722
0-1-5none012870
6-0-0none01320
0-6-0all6132792
0-0-6none01320
Successful total1294604
Possible total62509711505826
Successful percent85.9730141463887594

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

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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-ENorth SuccessFactorRatioProduct
3-2-2all105121681277640
2-3-2all105121681277640
2-2-3all105121681277640
3-3-1all1057436780780
3-1-3all1057436780780
1-3-3all1057436780780
4-2-1J-x, 10-x + J East355070177450
4-1-2J15507076050
2-4-1all1055070532350
2-1-4J15507076050
1-4-2all1055070532350
1-2-4J-x, 10-x + J West355070177450
5-1-1none015210
1-5-1all1051521159705
1-1-5none015210
4-3-0J-x-x45143064350
4-0-3none014300
3-4-0all1051430150150
3-0-4none014300
0-4-3all1051430150150
0-3-4J-x-x, 10-9-x57143081510
5-2-0J-1057023510
5-0-2none07020
2-5-0all10570273710
2-0-5none07020
0-5-2all10570273710
0-2-5J-1057023510
6-1-0none01560
6-0-1none01560
1-6-0all10515616380
1-0-6none01560
0-6-1all10515616380
0-1-6none01560
7-0-0none0120
0-7-0all105121260
0-0-7none0120
Successful total8541285
Possible total10510755911293695
Successful percent75.6287911086672697

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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-ENorth SuccessFactorRatioProduct
3-3-2all8404461637477440
3-2-3all but 10-9 + J West8254461636808200
2-3-3all8404461637477440
4-2-2J-x, 10-9 + J East220304206692400
2-4-2all8403042025552800
2-2-4J-x, 10-9 + J East230304206996600
4-3-1J-x-x, 10-x-x + J East360185906692400
4-1-3J105185901951950
3-4-1all8401859015615600
3-1-4J105185901951950
1-4-3all8401859015615600
1-3-4J-x-x, 10-x-x + J West360185906692400
5-2-1J-10, J-9 + 10 East359126319410
5-1-2none091260
2-5-1all84091267665840
2-1-5none091260
1-5-2all84091267665840
1-2-5J-10, J-9 + 10 West359126319410
4-4-0J-x-x-x42035751501500
4-0-4none035750
0-4-4J-x-x-x, 10-9-x-x, 10-8-7-x, 9-8-7-x61235752187900
5-3-0J-10-x, J-9-x1652574424710
5-0-3none025740
3-5-0all84025742162160
3-0-5none025740
0-5-3all84025742162160
0-3-5J-10-x902574231660
6-1-1none020280
1-6-1all84020281703520
1-1-6none020280
6-2-0J-103093628080
6-0-2none09360
2-6-0all840936786240
2-0-6none09360
0-6-2all840936786240
0-2-6J-103093628080
7-1-0none01560
7-0-1none01560
1-7-0all840156131040
1-0-7none01560
0-7-1all840156131040
0-1-7none01560
8-0-0none090
0-8-0all84097560
0-0-8none090
Successful total227767170
Possible total840430236361398240
Successful percent63.0238736082389333


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.

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© 2019 Richard Pavlicek