Main   Study 8J33 by Richard Pavlicek  

Fit To Be Tied

Did you ever wonder what’s the chance of a 4-4 fit? (Or any length for that matter.) Or which suit fits are most common? Or how often you’ll have a major fit? Or a dual fit? I could keep asking questions but expect the popular answer would be: Who gives a ……! Alas, that’s probably correct, but I lured you onto this page, so pay attention — or I’ll take away your masterpoints.
Suit FitsMajor FitsDual FitsDual Major FitsReciprocal Dual FitsLopsided Dual Fits

The probability of various suit fit lengths is not a simple calculation, as there are 635+ billion (52c13) bridge hands, each of which can face 8+ billion (39c13) others, producing 5+ sextillion (22 digits!) different combinations. Now there’s a sexy number! Nonetheless, with efficient algorithms and the aid of a computer it is readily obtained.

Percents are shown to four decimal places unless exact, so 0.0000 is greater than zero but simply rounds that way to the nearest 10,000th.

Suit Fits

The following table lists all possible suit fits in order of frequency. Columns show the percent chance that a side (two hands) has that fit in a specific suit; at least that fit in a specific suit; has that fit in any suit; at least that fit in any suit; has that fit as its longest fit; shortest fit; only fit of seven or more cards; and only fit of eight or more cards. Fit lengths for “at least” and “longest” are judged first by the combined length, and second by the longer length (e.g., a 5-3 fit is longer than a 4-4 fit); ties (e.g., two 6-2 fits) are credited as if there were no tie.

Every pair of hands has four “fits” as the term is used here, which does not necessarily imply a desirable trump suit — except for sadists who enjoy passing splinter bids — and each existing fit is acknowledged in the table. Percents indicate the portion of sides that contain the fit, not the number of occurrences of that fit; e.g., if a side has two (or three) 4-4 fits, it is counted only once in the 4-4 category.

The order of unbalanced fits (e.g., 5-3 versus 3-5) is ignored; either is counted the same. The percent chance for a specified order would be slightly more than half the value shown — not exactly half, because sides with the fit both ways would be counted for either order.

One revelation of practical benefit is that you will have at least a 4-4 fit over 84 percent of the time.

Note the curious entry of exactly 50 percent for at least a 4-3 fit in a specific suit — obvious if you think about it, as every fit must be either above or below the mean of 6.5 cards, and logically half would go each way.

Another curiosity is the anomaly regarding 4-2 and 3-2 fits. For a specific suit, sides with a 4-2 fit are more common; but for any suit, sides with a 3-2 fit have the edge. Go figure. (In all other cases the frequency order is the same in both columns.) This prompted me to recheck my program, but I’m confident the numbers are right.

Suit FitSpecific at LeastAny Suit at LeastLongestShortestOnly 7+Only 8+
4-314.82795046.91711005.754300
4-211.554265.376038.726610005.4857
3-210.976590.032538.9743100033.2158
3-38.473173.849129.633410009.8658
5-37.625820.855228.128871.457322.272402.471716.6840
5-27.279235.172125.918894.24577.590100
4-45.295726.150919.927884.263712.806401.716511.5861
4-14.573679.056017.4153100011.1170
5-44.33289.967516.959738.519116.353803.862011.9080
3-13.658894.395014.3962100013.8278
5-13.466353.821812.979210000.3778
2-22.993697.388611.8118100011.6661
6-22.773013.229410.671649.18499.008000.89886.0669
6-32.31085.63479.141122.16538.956502.05986.3509
2-11.696499.56346.771610006.7687
6-11.617627.89296.168086.65562.205000
6-41.00471.93334.01387.71904.008101.58183.2035
4-00.703690.73612.803910002.6060
5-50.67822.61142.710010.41922.700201.06772.1623
5-00.633374.48242.505610001.4118
7-20.63023.32382.504113.20882.466300.56181.7321
3-00.478597.86701.912710001.9042
7-10.462210.45631.815740.17691.565700.14981.0112
7-30.40190.92851.60633.71091.604800.63271.2814
6-00.355550.35551.399310000.0070
6-50.22610.43660.90421.74630.904200.50830.7930
1-10.205699.95880.822510000.8225
2-00.189899.75320.759210000.7591
7-40.12560.21050.50230.84210.502300.28240.4405
7-00.124426.27530.491684.45070.186900
8-20.08220.52670.32862.10610.328400.12940.2621
8-10.07882.69360.314110.74250.310000.07020.2165
1-00.039599.99840.158210000.1582
8-30.03770.08500.15070.33980.150700.08470.1322
8-00.02679.99420.106138.61120.092100.00860.0583
7-50.01810.02910.07230.11650.072300.05110.0673
6-60.01210.04120.04820.16470.048200.03410.0449
9-10.00760.44450.03041.77770.030400.01200.0243
8-40.00750.01100.03010.04420.030100.02130.0281
9-20.00570.04730.02280.18910.022800.01280.0200
9-00.00342.61480.013510.43250.013300.00300.0093
9-30.00170.00350.00670.01400.006700.00470.0062
0-00.00161000.006610000.0066
7-60.00090.00160.00370.00660.003700.00300.0036
8-50.00050.00070.00210.00280.002100.00170.0020
10-10.00040.04160.00150.16630.001500.00090.0013
10-00.00020.43680.00091.74720.000900.00040.0007
10-20.00020.00180.00070.00730.000700.00050.0007
9-40.00020.00020.00060.00080.000600.00050.0006
10-30.00000.00000.00010.00010.000100.00010.0001
11-10.00000.00160.00000.00660.000000.00000.0000
11-00.00000.04120.00000.16480.000000.00000.0000
11-20.00000.00000.00000.00000.000000.00000.0000
12-00.00000.00160.00000.00660.000000.00000.0000
12-10.00000.00000.00000.00000.000000.00000.0000
13-00.00000.00000.00000.00000.000000.00000.0000
56100Specific sideprints: 239344  Sides: 5157850293780050462400

From the above table you can also derive the expected number of occurrences of any fit. For example, suppose you wanted to know how many 4-4 fits to expect in 10000 deals. The chance of a 4-4 fit in a specific suit (from the table) is 5.2957 percent, which must be the same for each suit, so the answer is 4 × 5.2957% × 10000 = ~2118 for a given side; or twice that for both sides. TopMain

Major Fits

The next table is like the previous but concerns only the major suits. The specific column (including its at least) is identical but repeated for convenience. Next columns show the percent chance that a side (two hands) has that fit in either major; at least that fit in either major; has that fit as its longer major fit; shorter major fit; only major fit of seven or more cards; and only major fit of eight or more cards.

Observe that there are no exact zeros in the Longer and Shorter columns, as it’s possible for your longer major fit to be 0-0, or your shorter major fit to be 13-0 — and Mars could crash into Venus for that matter.

The table would also apply to any two designated suits, e.g., both minors, or hearts plus diamonds. Note that it would not apply to indefinite specifications like spades plus a minor, or major plus minor; these could be calculated, but I think I’ve gone too far already.

MajorSpecific at LeastEither Maj at LeastLongerShorterOnly 7+Only 8+
4-314.82795027.337880.627918.489911.166016.1718
4-211.554265.376021.760292.19378.285814.8227
3-210.976590.032521.123799.77481.792320.1608
3-38.473173.849116.189396.58104.387412.5588
5-37.625820.855214.853639.706413.87031.38139.668012.9107
5-27.279235.172114.002062.138010.58153.97697.9389
4-45.295726.150910.380848.88279.17641.41506.71398.9657
4-14.573679.05609.000197.98261.21177.9354
5-44.33289.96758.603719.70998.50280.16296.20977.7578
3-13.658894.39507.277899.94420.13467.1831
5-13.466353.82186.781983.90792.96053.9721
2-22.993697.38865.960199.99560.05145.9358
6-22.773013.22945.476025.83615.19820.34793.51564.6948
6-32.31085.63474.604711.20714.57390.04783.31184.1375
2-11.696499.56343.3904100.00000.00283.3899
6-11.617627.89293.184651.55662.47930.75591.7642
6-41.00471.93332.00863.86422.00760.00181.58771.8734
4-00.703690.73611.405599.80960.03471.3725
5-50.67822.61141.35595.21851.35430.00211.07171.2646
5-00.633374.48241.261996.77090.18981.0767
7-20.63023.32381.25776.63321.25140.00910.90321.1284
3-00.478597.86700.956799.99720.00160.9553
7-10.462210.45630.918820.63790.87710.04720.58590.7825
7-30.40190.92850.80361.85650.80330.00050.63510.7494
6-00.355550.35550.707280.94740.31950.3915
6-50.22610.43660.45210.87320.45210.00000.38540.4336
1-10.205699.95880.4112100.00000.00000.4112
2-00.189899.75320.3796100.00000.00000.3796
7-40.12560.21050.25120.42110.25120.00000.21410.2409
7-00.124426.27530.247849.07730.19450.05430.1357
8-20.08220.52670.16441.05320.16430.00010.12990.1533
8-10.07882.69360.15745.38190.15670.00080.11290.1411
1-00.039599.99840.0791100.00000.00000.0791
8-30.03770.08500.07540.16990.07540.00000.06420.0723
8-00.02679.99420.053219.76080.05090.00240.03380.0451
7-50.01810.02910.03620.05820.03620.00000.03260.0353
6-60.01210.04120.02410.08240.02410.00000.02180.0236
9-10.00760.44450.01520.88890.01520.00000.01200.0142
8-40.00750.01100.01510.02210.01510.00000.01360.0147
9-20.00570.04730.01140.09460.01140.00000.00970.0109
9-00.00342.61480.00675.22520.00670.00000.00480.0060
9-30.00170.00350.00330.00700.00330.00000.00300.0033
0-00.00161000.00331000.00000.0033
7-60.00090.00160.00190.00330.00190.00000.00170.0018
8-50.00050.00070.00100.00140.00100.00000.00100.0010
10-10.00040.04160.00080.08310.00080.00000.00060.0007
10-00.00020.43680.00050.87370.00050.00000.00040.0004
10-20.00020.00180.00040.00370.00040.00000.00030.0004
9-40.00020.00020.00030.00040.00030.00000.00030.0003
10-30.00000.00000.00010.00010.00010.00000.00000.0001
11-10.00000.00160.00000.00330.00000.00000.00000.0000
11-00.00000.04120.00000.08240.00000.00000.00000.0000
11-20.00000.00000.00000.00000.00000.00000.00000.0000
12-00.00000.00160.00000.00330.00000.00000.00000.0000
12-10.00000.00000.00000.00000.00000.00000.00000.0000
13-00.00000.00000.00000.00000.00000.00000.00000.0000
56100Specific sideprints: 239344  Sides: 5157850293780050462400
TopMain

Dual Fits

Every side (two hands) has four “fits” as defined above, but now let’s consider fits in the more usual way as in choosing a trump suit. To that extent, many sides have dual fits — two suits, either of which might be a viable contract. For practical purposes, only fits of 7-11 cards will be considered, with no hand having more than seven cards. The latter stems from the old bridge maxim: What do you call an eight-card suit? If you don’t know the answer, here’s a hint: White House travesty.

The following table shows the percent chance of a side’s two longest fits (7+ cards each) being as denoted by each column/row intersection. For example, the chance of a 5-3 fit and 4-4 fit being the two longest is found in the 5-3 column at the 4-4 row (or vice versa) to be 3.2518 percent. The unlikely presence of a third fit is ignored; only the two longest fits are considered.

Suit Fit4-35-26-17-04-45-36-27-15-46-37-25-56-47-36-57-4
4-35.75435.45130.88840.04416.21898.34352.46140.29205.21342.42080.50020.74451.02610.32930.19460.0921
5-25.45132.13871.08800.08303.08414.76881.99220.36092.40191.46810.46290.30460.49440.23700.07840.0524
6-10.88841.08800.22850.05520.54001.03100.64910.18240.41070.37270.18520.04390.09560.07520.01120.0128
7-00.04410.08300.05520.00470.02670.06900.06540.02610.01990.02950.02200.00160.00550.00710.00040.0009
4-46.21893.08410.54000.02671.22033.25180.92300.09891.47460.65340.12000.16510.21990.06280.03480.0149
5-38.34354.76881.03100.06903.25182.33661.59970.23051.97301.01470.24880.21130.30820.11500.04450.0238
6-22.46141.99220.64910.06540.92301.59970.41830.19230.56430.42430.17120.05280.10070.06380.01110.0102
7-10.29200.36090.18240.02610.09890.23050.19230.03290.06050.07640.04850.00430.01300.01440.00090.0017
5-45.21342.40190.41070.01991.47461.97300.56430.06050.37170.33040.06070.06960.09270.02650.01220.0052
6-32.42081.46810.37270.02950.65341.01470.42430.07640.33040.10230.06410.02780.04640.02270.00490.0035
7-20.50020.46290.18520.02200.12000.24880.17120.04850.06070.06410.01690.00400.01040.00950.00070.0011
5-50.74450.30460.04390.00160.16510.21130.05280.00430.06960.02780.00400.00280.00700.00160.00080.0003
6-41.02610.49440.09560.00550.21990.30820.10070.01300.09270.04640.01040.00700.00500.00350.00100.0005
7-30.32930.23700.07520.00710.06280.11500.06380.01440.02650.02270.00950.00160.00350.00130.00020.0003
6-50.19460.07840.01120.00040.03480.04450.01110.00090.01220.00490.00070.00080.00100.00020.00000.0000
7-40.09210.05240.01280.00090.01490.02380.01020.00170.00520.00350.00110.00030.00050.00030.00000.0000
TopMain

Dual Major Fits

The next table is like the previous but concerns only the major suits (or any two designated suits). Each column/row intersection shows the percent chance of a side having that dual major fit (7+ cards each). The previous stipulation of being the two longest fits (ignoring a third fit) no longer applies, as only two suits are involved.

For example, the chance of a 6-2 fit in one major and a 5-3 fit in the other is found in the 6-2 column at the 5-3 row (or vice versa) to be 0.2728 percent. It might seem curious that this is more likely than two 4-4 fits (0.2106 percent), but this is because there are two ways to comprise the former (either major could be the 6-2) but only one way if both are 4-4. Hence the chance of specifically 6-2 spades and 5-3 hearts could be found by halving the value (0.1364 percent).

Major4-35-26-17-04-45-36-27-15-46-37-25-56-47-36-57-4
4-32.31812.08620.38630.02211.49782.05410.64190.08151.02970.49040.10510.13220.18360.05990.03300.0157
5-22.08620.55630.27820.02350.64190.97570.40230.07580.44130.26480.08410.05290.08500.04050.01320.0088
6-10.38630.27820.05060.01220.10700.19970.12270.03430.07360.06540.03220.00760.01630.01270.00190.0021
7-00.02210.02350.01220.00100.00510.01300.01230.00490.00350.00510.00380.00030.00090.00120.00010.0001
4-41.49780.64190.10700.00510.21060.56170.16050.01720.24780.11010.02020.02750.03670.01050.00580.0025
5-32.05410.97570.19970.01300.56170.39800.27280.03970.33040.16990.04180.03520.05140.01920.00740.0040
6-20.64190.40230.12270.01230.16050.27280.07010.03220.09440.07080.02850.00880.01680.01060.00190.0017
7-10.08150.07580.03430.00490.01720.03970.03220.00550.01010.01270.00810.00070.00220.00240.00020.0003
5-41.02970.44130.07360.00350.24780.33040.09440.01010.06200.05510.01010.01160.01550.00440.00200.0009
6-30.49040.26480.06540.00510.11010.16990.07080.01270.05510.01700.01070.00460.00770.00380.00080.0006
7-20.10510.08410.03220.00380.02020.04180.02850.00810.01010.01070.00280.00070.00170.00160.00010.0002
5-50.13220.05290.00760.00030.02750.03520.00880.00070.01160.00460.00070.00050.00120.00030.00010.0000
6-40.18360.08500.01630.00090.03670.05140.01680.00220.01550.00770.00170.00120.00080.00060.00020.0001
7-30.05990.04050.01270.00120.01050.01920.01060.00240.00440.00380.00160.00030.00060.00020.00000.0000
6-50.03300.01320.00190.00010.00580.00740.00190.00020.00200.00080.00010.00010.00020.00000.00000.0000
7-40.01570.00880.00210.00010.00250.00400.00170.00030.00090.00060.00020.00000.00010.00000.00000.0000
TopMain

Reciprocal Dual Fits

Most dual fits are reciprocal when the suit lengths of each are unequal. For example, two 5-3 fits are more than twice as likely to occur with the 5-card lengths in opposite hands, as opposed to one hand being 5-5. In extreme cases this is forced; e.g., dual fits of 7-3 and 7-2 would have to be reciprocal, else a hand has 14 cards.

The following table is like the one for Dual Fits but only reciprocal dual fits are counted. Note that equal-length fits (4-4 and 5-5) have been removed, as they have no relevance in this issue.

Suit Fit3-42-51-60-73-52-61-74-53-62-74-63-75-64-7
4-32.93213.51640.64500.03484.97511.68420.22212.86731.56410.36720.62040.23180.10840.0616
5-23.51641.82621.03960.08203.65971.82990.35251.55821.27150.44520.38920.22240.05250.0471
6-10.64501.03960.22680.05510.89850.63760.18220.30200.35780.18460.08680.07470.00880.0126
7-00.03480.08200.05510.00470.06500.06520.02610.01640.02930.02200.00530.00710.00040.0009
5-34.97513.65970.89850.06501.59141.32280.21291.18610.79070.22480.22030.10060.02780.0199
6-21.68421.82990.63760.06521.32280.40300.19140.39560.39630.16980.08810.06290.00830.0099
7-10.22210.35250.18220.02610.21290.19140.03290.04850.07550.04850.01260.01440.00080.0017
5-42.86731.55820.30200.01641.18610.39560.04850.20650.22030.04720.05800.01990.00700.0037
6-31.56411.27150.35780.02930.79070.39630.07550.22030.09180.06290.03860.02210.00350.0033
7-20.36720.44520.18460.02200.22480.16980.04850.04720.06290.01690.00990.00950.00060.0011
6-40.62040.38920.08680.00530.22030.08810.01260.05800.03860.00990.00390.00330.00070.0005
7-30.23180.22240.07470.00710.10060.06290.01440.01990.02210.00950.00330.00130.00020.0003
6-50.10840.05250.00880.00040.02780.00830.00080.00700.00350.00060.00070.00020.00000.0000
7-40.06160.04710.01260.00090.01990.00990.00170.00370.00330.00110.00050.00030.00000.0000
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Lopsided Dual Fits

The opposite case of a reciprocal dual fit is the lopsided variety, where one hand holds the greater length in both fits. For example, fits of 6-3 and 5-4 would be lopsided if one hand is 6-5. As obviously expected, lopsided dual fits are always less likely than reciprocal ones.

The following table shows the percent chance of lopsided dual fits. Impossible cases are grayed out.

Suit Fit4-35-26-17-05-36-27-15-46-37-26-47-36-57-4
4-32.82221.93500.24340.00933.36830.77720.06992.34610.85670.13310.40570.09740.08620.0305
5-21.93500.31250.04850.00101.10910.16240.00840.84370.19670.01770.10510.01460.02590.0052
6-10.24340.04850.00170.00000.13250.01150.00030.10870.01500.00060.00880.00050.00250.0002
7-00.00930.00100.00000.00410.00020.00360.00020.00010.0000
5-33.36831.10910.13250.00410.74520.27690.01760.78690.22400.02400.08800.01440.01670.0040
6-20.77720.16240.01150.00020.27690.01540.00090.16870.02800.00130.01260.00090.00280.0003
7-10.06990.00840.00030.01760.00090.01210.00090.00040.0001
5-42.34610.84370.10870.00360.78690.16870.01210.16520.11010.01350.03480.00660.00520.0015
6-30.85670.19670.01500.00020.22400.02800.00090.11010.01050.00110.00770.00060.00140.0002
7-20.13310.01770.00060.02400.00130.01350.00110.00050.0001
6-40.40570.10510.00880.00010.08800.01260.00040.03480.00770.00050.00120.00020.00030.0000
7-30.09740.01460.00050.01440.00090.00660.00060.00020.0000
6-50.08620.02590.00250.00000.01670.00280.00010.00520.00140.00010.00030.00000.00000.0000
7-40.03050.00520.00020.00400.00030.00150.00020.00000.0000
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© 2018 Richard Pavlicek