Main   Study 8J69 by Richard Pavlicek  

LoTT Variance by HCP

The so-called Law of Total Tricks (LoTT) is purported to apply regardless of the division of high cards, because a deficiency for one side is offset by an abundance for the other — or so it may seem. This study was inspired by Walt Schafer Jr. of Fountain Valley, California, who correctly notes that the LoTT becomes progressively less accurate as the high-card disparity increases. I thought it would be interesting to break down each total-trump situation according to the HCP apportionment between the two sides.

Total Trumps 14151617181920212223

Statistics are obtained from my database of 10,485,760 random deals. Each deal is double-dummy solved 20 times (four players × five strains), a project that itself took almost two years to complete. Actual play would not be double-dummy, of course, but previous studies have shown a close correlation between the two, so analyses are usually on track.

Law of Total Tricks

The sum of the lengths of each side’s longest fit is equal to the sum of the tricks each side can win with its longest fit trumps. Or to state it mathematically:

NS longest fit length + EW longest fit length  =  NS winnable tricks + EW winnable tricks

While hardly a law, the LoTT predicts a condition that is true on average for deals with 14-19 total trumps, but the variance is such that most deals do not produce the equality. As the total number of trumps increases, the percent of equalities decreases; and with 20+ total trumps the LoTT fails outright, as even the average number of total tricks is then below the number of total trumps. The high variance and failures suggest that judgment is more useful in practice than blindly following any law. For an amusing aspect see The Law of Total Trash.

There are at least four ways to calculate total tricks on a bridge deal: (1) Longest fit declared by longer or stronger hand, (2) longest fit right-sided whenever the declarer matters, (3) best fit declared by longer or stronger hand, or (4) best fit right-sided whenever the declarer matters. Note that a side’s longest fit is not necessarily its best fit, e.g., a 4-4 fit might produce more tricks than a 6-3 fit. Practical considerations suggest Option 1, which is used in this study. Further, in the event of a tie (e.g., one side with two 8-card fits) the more productive fit is chosen.

The following tables show the LoTT behavior for 14-23 total trumps (deals with 24-26 total trumps were too rare to provide meaningful statistics). Column 1 shows the HCP apportionment between the two sides, Column 2 shows the number of cases, and Column 3 shows the average number of total tricks. The remaining columns show the percent occurrence of each number of total tricks. The highest percent in each row is tinted gold, the second highest a lighter gold, and the third highest lighter still.

14 Total Trumps

10-20 Total Tricks
HCPCasesAvg1011121314151617181920
20-208734714.2900.01260.326310.707953.762630.49684.28750.37670.02750.00230
21-1917260514.290.00060.01740.354010.665453.471230.71464.36020.38350.03130.00170
22-1816263314.300.00060.01290.333310.850852.674431.20154.48370.41380.02640.00250
23-1714787014.3100.01560.332010.743952.477931.32014.65810.41590.03380.00200.0007
24-1612908814.320.00080.01700.400510.478151.396732.18585.05550.42840.03410.00310
25-1510787414.350.00190.01020.413410.467849.821133.25275.51110.48300.03520.00370
26-148657414.370.00230.02430.419310.237548.363234.34636.02490.54400.03470.00350
27-136663914.390.00600.01650.451710.264346.920034.88656.75430.66180.03600.00300
28-124862914.4200.02060.547010.018745.719635.33287.60660.70740.04320.00410
29-113441414.450.00870.04650.54639.836143.444536.58398.63310.85430.046500
30-102301714.470.01300.04780.56059.836242.221037.27689.05850.91670.06520.00430
31-91479714.480.02030.03380.621710.062941.021837.72399.43431.02720.04050.00680.0068
32-8865014.5100.08090.39319.687940.612738.069410.00001.12140.034700
33-7498114.4800.02010.72279.877541.437537.38219.47601.04400.040200
34-6250914.49000.63779.804740.374738.82029.52570.79710.039900
35-5122514.4200.08161.224511.428642.122435.34698.97960.8163000
36-455314.28001.446713.562447.016332.36894.88250.7233000
37-318614.33001.07539.677453.225828.49466.45161.0753000
38-27514.0900016.000060.000022.66671.33330000
39-12014.0000020.000060.000020.000000000
40-0113.000001000000000
All109968714.340.00180.01830.394810.516950.581832.54215.41210.49570.03370.00260.0002

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15 Total Trumps

10-21 Total Tricks
HCPCasesAvg101112131415161718192021
20-208891215.210.00110.00340.03260.976214.706750.921128.87804.12430.33290.02140.00220
21-1917285915.2200.00230.04171.028014.762950.261229.38874.15310.34130.02020.00060
22-1816328015.2200.00120.04231.054014.816350.072929.38274.25890.34660.02390.00060.0006
23-1714828815.230.00070.00200.04251.111414.899449.389729.61264.52300.39520.02090.00270
24-1612924415.2300.00460.06191.231014.887348.690129.80254.86680.42400.02860.00310
25-1510709515.240.00090.00560.07561.297015.325647.475630.13965.23270.41550.031700
26-148646815.2500.00690.07291.401715.371046.698230.11635.74090.55860.033500
27-136606015.260.00150.00300.09541.521315.644945.752330.14836.20040.58280.050000
28-124854615.250.00210.02060.09481.664416.007544.856430.34246.39810.56240.04740.00410
29-113428215.260.00580.00880.09631.995216.148443.985230.21126.84320.65050.05250.00290
30-102269215.2500.01320.15422.124117.028042.878529.69777.34180.72270.039700
31-91425215.230.00700.01400.13332.147117.120443.748229.28016.87620.62450.049100
32-8849415.1900.02350.23552.672518.330643.454227.74906.81660.68280.035300
33-7478815.1700.02090.27152.798718.964143.316627.65256.32830.58480.062700
34-6243115.0900.04110.37022.427022.130844.261624.76355.63550.3702000
35-5112014.95000.35714.107124.017946.785721.51792.76790.4464000
36-449614.84000.40326.653226.008146.371017.13713.22580.2016000
37-318614.730003.763431.182854.30119.67741.07530000
38-26214.500004.838743.548448.38713.225800000
39-12114.52000057.142933.33339.523800000
40-0214.0000001000000000
All109957815.230.00070.00490.06381.273215.212448.389729.62484.96800.43190.02910.00140.0001

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16 Total Trumps

9-22 Total Tricks
HCPCasesAvg910111213141516171819202122
20-2022816616.16000.00180.02410.28533.247619.450842.605426.78096.47730.98880.12710.01100
21-1944871916.1600.00020.00290.02990.30493.264019.691242.572726.60226.40620.99640.11920.00960.0007
22-1842158316.1500.00050.00550.02400.32213.327719.840542.287826.69566.37240.99410.12380.00620
23-1738196716.1400.00030.00550.03800.35953.606120.183442.065226.37666.26050.97630.11990.00860.0003
24-1633105716.1300.00120.00660.04200.40573.741720.435841.729126.33476.23670.94760.10900.00970.0003
25-1527764916.1000.00180.01080.06010.51324.151321.021141.398025.79266.03640.90400.10370.00680
26-1422035716.070.00050.00090.01040.06760.57594.593521.511541.021225.48825.85640.79370.07620.00320.0009
27-1316981616.0300.00470.01770.09010.71255.098522.354240.540924.78395.57490.73370.07830.01000.0006
28-1212283115.990.00080.00980.02040.13190.86385.627223.490040.035523.77905.35860.62280.05450.00570
29-118630615.9400.00460.02550.12981.10776.370424.335539.538422.92544.96140.56540.03010.00460.0012
30-105620215.8700.00530.04450.18681.28827.357425.762438.587621.61844.69910.42350.026700
31-93572415.8000.00560.02800.19871.48648.428527.368237.957720.09574.10930.31350.008400
32-82088015.6900.00480.05750.27301.72899.363030.407137.140817.72033.02200.27300.009600
33-71155215.5900.00870.06930.34631.904411.037032.003136.669015.52982.29400.1385000
34-6570415.44000.05260.42082.717412.219536.974133.643112.23701.68300.0526000
35-5268015.35000.03730.37312.761212.985140.261233.95528.50751.04480.0746000
36-4113315.230000.52963.707014.033543.424531.15626.97260.17650000
37-346815.060000.42742.991516.453052.350425.64102.136800000
38-213214.8900004.545516.666765.151512.12121.515200000
39-13414.8800002.941220.588261.764714.7059000000
40-0614.670000033.333366.66670000000
All282296616.100.00010.00160.00960.05780.50104.132221.008141.565625.70526.02480.88450.10160.00750.0003

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17 Total Trumps

9-23 Total Tricks
HCPCasesAvg91011121314151617181920212223
20-2019686917.13000.00050.00100.01630.29463.808621.660639.978426.71476.61510.83460.07310.00250
21-1938786917.1200.00030.00080.00230.02140.30963.874521.451140.342026.49146.58240.85260.06570.00570.0003
22-1836396717.120000.00190.02880.32703.928921.567140.079226.55496.57890.84540.08380.00380.0003
23-1732886917.1100.00030.00060.00610.02740.35584.111121.940739.688826.32176.57440.89060.07540.00730
24-1628475017.1000.00040.00140.00840.03690.42774.368722.129939.312026.01766.73850.88110.07270.00460
25-1523505217.0800.00090.00170.00720.06510.51014.815522.571238.857825.56296.63560.88410.08250.00510.0004
26-1418794417.0500.00210.00210.01490.08940.60765.341523.294738.090124.96916.63500.87690.07240.00370.0005
27-1314210117.01000.00210.01480.11190.75515.991524.129337.310124.35386.39970.83390.09080.00700
28-1210240116.9700.00490.01170.02830.11720.90626.728424.634537.018223.60046.09080.77340.08110.00490
29-117055716.910.00140.00140.01130.02410.15591.16227.864625.509935.766822.85245.90590.67320.070900
30-104629316.8400.00430.00220.03240.22251.51649.141826.336635.033421.62105.37230.65240.06260.00220
31-92837416.7400.00700.01760.05290.32072.008910.897327.891733.675219.88794.70500.48640.04230.00700
32-81652616.64000.03630.03630.25412.208612.997729.075433.305117.78413.92710.33890.03030.00610
33-7903816.53000.01110.04430.39832.832515.169330.460331.102016.28683.40780.27660.011100
34-6440116.37000.02270.11360.54533.726417.586933.310627.811914.33772.43130.1136000
35-5200616.200000.09970.39884.037922.083735.443725.324011.41581.09670.04990.049900
36-480116.110000.12480.24974.619225.218534.956323.470710.73660.62420000
37-329015.8400000.34484.827638.620730.344818.62077.241400000
38-27915.67000005.063346.835427.848116.45573.797500000
39-12115.71000004.761938.095242.85719.52384.761900000
40-0316.0000000001000000000
All240821117.070.00000.00080.00230.00920.05950.52804.879822.593738.978425.56226.46900.83760.07470.00480.0002

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18 Total Trumps

12-24 Total Tricks
HCPCasesAvg12131415161718192021222324
20-2013679618.07000.02050.50886.226123.164436.296424.97087.32481.29760.17400.01610.0007
21-1926775518.0700.00070.02170.50576.314423.194036.486024.59457.37051.32850.16880.01420.0011
22-1825176518.0600.00160.02140.52716.336923.404436.292224.52537.41681.29010.16600.01710.0012
23-1722602618.0600.00270.03140.51816.447523.694636.005124.26547.48281.36530.17080.01550.0009
24-1619434518.0500.00150.03140.55626.533724.009935.881623.98727.46351.34450.17960.01030.0005
25-1516025118.0200.00190.03000.61226.776224.825435.652823.23237.40901.29050.15790.01120.0006
26-1412589017.9900.00320.04050.68087.257125.386435.367422.50067.32781.29080.13820.00640.0008
27-139410017.950.00110.00430.05950.84807.604726.399635.119021.64086.99261.18380.13820.00850
28-126740217.9000.00150.06380.98228.440427.066035.168120.62106.55920.99110.10390.00300
29-114622617.8300.00870.07791.23529.354028.509934.526019.37446.10050.74420.06490.00430
30-102953017.7400.00680.12871.544210.921129.888233.261118.28315.27600.63330.05080.00680
31-91766017.6300.00570.16422.095112.938831.251432.191416.56854.26390.48700.034000
32-81009317.5300.00990.19822.576015.119431.467431.398015.21853.65600.30710.049500
33-7518117.3900.01930.21233.821716.772833.410530.611911.69663.26190.17370.019300
34-6255117.2200.03920.31365.448820.188234.065128.02829.95691.84240.07840.039200
35-5110717.03000.81307.226723.757934.688325.11296.95571.35500.0903000
36-445016.69000.444412.000033.777831.111117.33335.11110.22220000
37-315616.47000.641014.743642.307722.435918.58971.282100000
38-23816.052.631602.631623.684242.105318.42117.894702.63160000
39-1816.250012.500012.500050.0000012.500012.500000000
All163733018.010.00010.00230.03820.67727.007424.529335.763523.38087.18721.24680.15440.01210.0007

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19 Total Trumps

13-24 Total Tricks
HCPCasesAvg131415161718192021222324
20-207611318.8200.00130.05781.22459.834128.351334.200519.54595.65610.98800.13270.0079
21-1914826818.8200.00070.05731.163410.075728.364934.225919.41425.62700.96380.10180.0054
22-1813832318.8200.00580.05491.246410.086528.515133.563519.59335.80741.00270.12000.0043
23-1712426618.8100.00160.05231.252210.315828.472833.595719.28365.93810.96810.11270.0072
24-1610562118.8000.00380.05961.264910.696728.915733.176219.03035.75740.98180.09850.0151
25-158689518.7700.00350.08981.398211.040929.132932.954718.69385.62290.96320.09320.0069
26-146753118.7500.00150.10071.476411.554729.759732.268118.33085.45680.94480.09630.0104
27-135021818.7000.00400.09961.762312.539330.166531.685817.68095.07190.90800.07770.0040
28-123510318.640.00280.01140.12531.957113.545930.909031.390516.90174.34440.72360.08830
29-112355318.560.00850.01270.19952.284215.059731.647830.501415.64564.00370.58590.04670.0042
30-101465518.4900.04090.17062.736316.574532.419029.205014.81413.50050.49810.04090
31-9870118.3400.04600.34483.597319.158733.984627.376212.73422.44800.27580.03450
32-8476118.2200.04200.52514.557921.067035.349725.834910.50201.86940.252000
33-7256918.1100.19460.77856.228123.394333.047924.83469.88711.32350.311400
34-6119817.9900.16690.33398.430724.457435.058422.87157.51251.085100.08350
35-548417.7700.41321.033110.330627.892634.504122.10742.89260.8264000
36-417317.5300.57801.734118.497124.277535.260117.91911.73410000
37-35717.25001.754426.315831.578926.315814.035100000
38-21916.9505.2632031.578931.578921.052610.526300000
39-1316.00033.3333033.3333033.3333000000
All88851118.770.00030.00600.08251.450911.106629.124533.031218.69275.46900.92830.10120.0069

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20 Total Trumps

14-26 Total Tricks
HCPCasesAvg14151617181920212223242526
20-203245219.5400.00920.11402.662415.983631.119830.555914.71404.06750.70570.067800
21-196358319.5400.00310.12582.653215.917831.409330.266014.73044.14730.69040.05500.00160
22-185930019.5400.00510.15682.718415.394631.715030.089415.02194.11970.68970.07250.01520.0017
23-175270319.5200.00570.11762.929615.968732.062729.512614.47934.24830.61670.05690.00190
24-164509619.5100.00220.20842.958116.030232.047229.548114.57113.95160.59870.07760.00670
25-153624219.48000.23453.261416.682332.156128.762214.29563.90710.63190.06350.00550
26-142799319.4400.01790.28583.518717.211432.776127.985613.74273.77240.64660.03570.00710
27-132008519.3900.01990.28884.232017.854132.566628.001013.01973.48520.48790.044800
28-121406119.3200.01420.44804.829019.074033.326226.619712.11153.15060.36980.04270.01420
29-11921519.2500.03260.36905.328321.280532.783525.794911.31852.64790.37980.065100
30-10564019.1500.03550.46107.322722.322733.209223.262410.35462.81910.17730.035500
31-9322319.050.031000.86888.005024.201131.957824.44938.47041.76850.18620.062100
32-8167718.930.05960.05960.77528.586826.774034.048920.93027.21531.31190.2385000
33-783818.7500.23871.074013.245828.878331.264916.82587.51790.83530.1193000
34-636318.5400.27553.030315.151529.476631.955916.52893.03030.55100000
35-514818.3600.67572.702719.594631.081133.78388.78382.02701.35140000
36-46517.9501.53857.692326.153830.769229.23083.076901.53850000
37-31417.2907.142928.571421.428614.285728.57140000000
38-2518.0000040.000020.000040.00000000000
All37270319.480.00050.00940.21093.289516.657031.990929.125314.13863.89850.61390.05980.00540.0003

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21 Total Trumps

15-25 Total Tricks
HCPCasesAvg1516171819202122232425
20-201087420.1300.01840.41387.357022.475632.941024.52649.74802.26230.25750
21-192109720.1400.00950.35556.754522.742634.066523.99879.46112.27520.32710.0095
22-181950520.1200.01540.54357.054623.250433.217123.66579.83852.19430.22050
23-171712020.1100.02920.63676.980123.510533.165923.77349.43342.14950.32130
24-161460920.0900.00680.55457.673424.245332.623722.89009.40522.21100.37650.0137
25-151157920.0600.01730.88957.841824.794932.412122.36819.00772.35770.29360.0173
26-14899020.0100.02220.85658.331525.550632.513922.22478.02002.18020.30030
27-13621419.9600.01611.09439.237225.844932.072721.88617.96591.68970.19310
28-12417619.8400.07181.173411.278727.729931.393719.75577.27971.10150.21550
29-11264419.8000.11351.777610.665728.593032.072619.17556.42970.98340.18910
30-10160219.6600.06243.058712.858929.525632.147315.79285.49310.74910.31210
31-987419.5600.34322.517215.675129.290631.922215.78953.54690.80090.11440
32-847519.4400.42114.421114.736834.105330.736810.10534.42110.63160.42110
33-720419.36004.411818.627531.372532.35299.31373.43140.490200
34-68118.9902.46913.703730.864232.098822.22223.70374.9383000
35-53218.723.125009.375028.125034.375018.75006.25000000
36-41118.6400054.545527.272718.181800000
All12008720.060.00080.02660.72207.821024.171632.936122.90269.03352.09350.28730.0050

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22 Total Trumps

16-26 Total Tricks
HCPCasesAvg1617181920212223242526
20-20281520.7500.07102.060410.479629.698033.179418.43694.72471.24330.07100.0355
21-19549120.7100.05462.513211.600829.211432.708117.61065.40890.85590.03640
22-18512320.7000.07812.440012.063229.377332.481017.21655.40700.81980.11710
23-17441820.690.02260.04532.354013.060228.519732.254417.81354.68541.19960.04530
24-16365220.6700.02742.081113.554230.011031.599116.18295.39431.04050.10950
25-15285520.6200.17512.486913.660230.472931.733815.37655.60420.38530.10510
26-14219020.6000.13702.694115.251129.908729.360717.39734.20091.00460.04570
27-13149320.4600.20093.282017.950431.480227.997314.80243.75080.46890.06700
28-1291020.4600.10992.637420.549529.340728.571413.73634.61540.32970.10990
29-1156120.2400.17835.704122.103433.333322.994712.29952.85200.534800
30-1035220.2600.28414.829522.159132.386425.284111.36363.6932000
31-916720.32006.586817.365334.730525.748510.77843.59281.197600
32-87619.8401.31585.263238.157925.000023.68426.57890000
33-74019.980010.000017.500047.500015.000010.00000000
34-61819.390027.777827.777827.777811.11115.55560000
35-5619.3300083.3333016.666700000
36-4219.000050.0000050.0000000000
All3016920.650.00330.08952.578813.517229.719231.449516.73574.95870.87180.07290.0033

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23 Total Trumps

17-25 Total Tricks
HCPCasesAvg171819202122232425
20-2059721.26004.522620.435537.018423.450611.72532.68010.1675
21-19106621.190.09380.28145.816121.013135.272024.671710.88181.87620.0938
22-1895621.2400.20924.497922.175732.949827.092110.77412.30130
23-1780621.27004.218420.471535.483926.302711.04222.23330.2481
24-1671221.2500.14045.056219.382036.236026.264010.11242.66850.1404
25-1550621.2700.39533.557322.332030.830031.02779.28852.37150.1976
26-1438421.0200.260410.937524.218829.687523.95838.33332.34380.2604
27-1326621.1200.75196.766922.932329.323330.07529.39850.75190
28-1216220.7500.617312.963030.864230.246917.90126.17281.23460
29-1112620.850011.904823.015938.888921.42863.96830.79370
30-104620.800010.869634.782632.60878.695710.86962.17390
31-92220.6804.5455040.909131.818222.7273000
32-8920.220033.333322.222233.333311.1111000
33-7319.670033.333366.666700000
All566121.190.01770.22965.741021.833634.039925.719810.13962.15510.1237

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© 2019 Richard Pavlicek