Main     Study 8J29 by Richard Pavlicek    

Bag of Tricks

Did you ever wonder how many tricks can be won on a bridge deal? I don’t mean a specific deal, which can be readily determined, but for deals in general. That is, given a random bridge deal, what is the probability that a certain number of tricks can be won? Or more precisely, what is the probability that a certain contract will be the highest makable?

Most TricksHighest ContractHighest Score
These answers are impossible to determine in theory, as it would require 20 double-dummy solutions (4 players × 5 strains) for every bridge deal, a task that would take billions of computer-years even by the fastest computers. While the number of bridge deals (53+ octillion) can be reduced to 558+ septillion distinct deals by equalizing suit permutations and hand rotations, it remains far out of reach. To bring this number into perspective, it has 27 digits, and the number of feet to the nearest star is only 18 digits. Mind-boggling.

Therefore, answers must be found empirically by creating and solving random deals. The larger the sample, the greater the probability the results will be accurate. This study is based on a database of 10,485,760 random deals, each double-dummy solved 20 times, a project that itself took two years to complete. The deals have been analyzed and show no apparent bias. For statistics see Random Solved Deals.

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. The highest percent in each column is tinted bright, and the second highest dim, to ease recognition.

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Most Tricks

The following table shows the percent of deals on which each number of tricks will be the most winnable in any strain. Columns show the percent for specific declarers, specific sides, and lastly for the whole deal. The difference between a specific declarer and the same side is that results for the side allow each contract to be played by the maximal declarer.

The table shows that 10 tricks is the most likely result of a bridge deal, with 9 tricks next. For a specific player or side, however, this drops to 9 tricks, with 8 tricks a close second.

TricksNorthSouthWestEastN-SW-EDeal
131.45401.45321.45551.45571.47431.47632.9506
125.06725.06775.06185.06275.31515.308210.6160
1110.304310.302910.306910.304610.496610.502520.7847
1015.246715.252815.242915.242115.382115.373828.3820
917.853917.849617.855617.855917.872117.876224.9277
817.252517.252317.249817.253117.156717.154811.0335
714.038814.037414.017414.017813.878113.85361.3054
69.63429.63409.65809.65609.47639.50230.0002
55.51665.51815.52195.52035.40395.40840
42.51522.51482.51752.51982.45682.45990
30.88690.88770.88100.88040.86440.85800
20.20850.20830.21100.21100.20280.20550
10.02100.02100.02040.02050.02050.02000
00.00020.00020.00030.00030.00020.00030
Average8.41058.41048.41008.41008.44208.44169.9996

The bottom line shows the average winnable tricks per deal. Curiously, the average for a deal (9.9996) is remarkably close to 10, which gives cause to wonder: Would a larger sample continue to approach 10? Could this be some mystical axiom of card play? Out of curiosity I checked two other sources, although small: GIB database averaged 9.9991 for 717,102 deals; Haglund database averaged 10.0011 for 100,000 deals. I might be on to something here! And in a billion years or so, I may be able to prove it.

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Highest Contract

The following table shows the percent of deals on which each contract will be the highest makable. Columns show the percent for specific declarers, specific sides, and lastly for the whole deal. The difference between a specific declarer and the same side is that results for the side allow each contract to be played by the maximal declarer.

In some cases no contract is makable, so in order to be complete, the table also includes subcontracts (below 1 C) indicated as “strain tricks”; e.g., H 6 = hearts winning six tricks.

The table shows that 4 S is the most likely highest makable contract on a bridge deal, with 3 S second. For a specific player or side, however, this drops to 3 S, with 2 S a close second. The spade suit rules, of course, because it wins all ties with other suits.

ContractNorthSouthWestEastN-SW-EDeal
7 NT0.68960.68980.68750.68740.69520.69311.3883
7 S0.23420.23420.23530.23560.24050.24160.4821
7 H0.20400.20380.20480.20460.20840.20930.4177
7 D0.17720.17600.17810.17770.17950.18080.3602
7 C0.14910.14950.14980.15030.15080.15150.3023
6 NT1.54671.54731.53781.53931.72371.71523.4389
6 S1.13961.14011.14131.14061.15801.15892.3164
6 H0.95760.95760.95860.95920.97600.97651.9508
6 D0.78500.78590.78830.78810.80380.80561.6069
6 C0.63820.63670.63570.63550.65370.65211.3029
5 NT1.99001.99161.98951.98922.11202.11154.2232
5 S2.66232.66162.65632.65412.68342.67815.3408
5 H2.24662.24542.25862.26042.26522.27904.4911
5 D1.87931.87851.87441.87261.89501.88923.7165
5 C1.52621.52581.52811.52831.54101.54473.0132
4 NT2.16162.16382.16202.16192.25992.25754.5097
4 S4.13794.13474.11914.12064.14964.13207.9413
4 H3.51273.51413.52693.52243.52213.53406.4593
4 D2.96392.96742.96882.97102.97332.97885.2501
4 C2.47062.47292.46612.46612.47722.47164.2216
3 NT2.16942.16652.16842.17362.22952.23174.3646
3 S4.90624.90394.89824.89254.89544.88707.6470
3 H4.19084.19354.19854.20104.18264.19015.6063
3 D3.57103.56883.57413.57243.56033.56294.1696
3 C3.01653.01703.01633.01643.00443.00453.1401
2 NT1.91521.91561.91901.91841.94061.94693.2578
2 S4.80484.80384.79894.80224.77194.76863.7326
2 H4.09124.08944.09814.10034.05774.06852.1254
2 D3.47663.47823.48093.47823.44903.44791.2263
2 C2.96472.96542.95292.95402.93742.92290.6913
1 NT1.44221.44141.44401.43871.44721.44531.0566
1 S4.02034.02204.01904.01933.97343.97190.2152
1 H3.37003.36603.35903.36113.32663.31660.0283
1 D2.82502.82762.82792.83192.78602.79010.0043
1 C2.38122.38042.36762.36692.34502.32970.0008
NT 61.02741.02871.03051.02961.02211.02600.0002
S 62.81742.81662.82352.82422.77102.77960.0000
H 62.31732.31792.32262.32022.27622.28120.0000
D 61.90481.90451.90971.90911.86951.87420
C 61.56741.56621.57161.57291.53751.54130
NT 50.59890.59860.59430.59570.59060.58680
S 51.66571.66801.67291.67191.63321.63800
H 51.33901.34021.33741.33691.31121.30980
D 51.06531.06421.06671.06541.04061.04280
C 50.84760.84720.85060.85030.82820.83110
NT 40.26910.26950.26980.27080.26440.26500
S 40.79590.79660.80390.80420.77780.78600
H 40.61710.61600.61320.61280.60150.59810
D 40.47080.47080.47210.47270.46000.46100
C 40.36230.36180.35860.35930.35300.34980
NT 30.09570.09530.09390.09340.09290.09120
S 30.30260.30220.29890.29950.29490.29170
H 30.22020.22100.21880.21900.21510.21370
D 30.15720.15780.15730.15680.15310.15270
C 30.11120.11150.11220.11170.10840.10880
NT 20.02880.02880.02960.02940.02780.02880
S 20.07850.07860.07910.07930.07660.07710
H 20.05190.05160.05120.05100.05040.04980
D 20.03060.03070.03180.03190.02990.03120
C 20.01870.01850.01930.01930.01800.01880
NT 10.00600.00600.00560.00580.00580.00550
S 10.00840.00840.00840.00830.00820.00820
H 10.00410.00410.00410.00410.00400.00400
D 10.00200.00190.00170.00170.00190.00170
C 10.00060.00070.00060.00060.00060.00060
00.00020.00020.00030.00030.00020.00030
Average2 NT2 NT2 NT2 NT2 NT2 NT4 H

The average highest contract, albeit meaningless, was found by encoding each contract as 0 (trickless) to 65 (7 NT), times its occurrences; summed, divided by 10485760, and decoded to the nearest contract.

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Highest Score

The following table shows the percent of deals on which each score will be the highest makable score. Columns show the percent for specific declarers, specific sides, and lastly for the whole deal. The difference between a specific declarer and the same side is that results for the side allow each contract to be played by the maximal declarer.

Scores are only for contracts made (not doubled or redoubled) and assume neither side is vulnerable. These are not the same as par scores; for that aspect see Par for the Course.

The table shows that 140 is the most common highest makable score on a bridge deal, followed closely by 420 (nonvul). For a specific player or side, however, the most common best score is zero (no makable contract) followed by the ubiquitous 110.

ScoreNorthSouthWestEastN-SW-EDeal
15200.68960.68980.68750.68740.69520.69311.3883
15100.43820.43790.44010.44020.44890.45090.8998
14400.32630.32560.32790.32810.33020.33230.6625
9901.54671.54731.53781.53931.72371.71523.4389
9802.09722.09772.09992.09982.13402.13544.2672
9201.42321.42271.42401.42361.45741.45762.9098
4601.99001.99161.98951.98922.11202.11154.2232
4504.90894.90704.91494.91454.94864.95719.8318
4303.04093.04183.03783.03633.18003.17386.3476
4208.40788.40778.40478.40218.43468.428716.1091
4005.17625.17395.16935.17195.25735.252810.0894
14010.361110.361410.356910.356910.343310.340416.5232
1302.93252.93602.94202.94292.91052.92083.6603
1203.33653.33633.33793.33853.37673.38135.9305
11014.062214.058314.068514.071213.958313.970110.4950
907.88357.88497.87797.87087.83377.81602.9743
807.39047.38807.37807.38037.30007.28850.2435
705.20625.20805.19555.19885.13095.11980.0052
018.782718.784218.810118.808218.425018.45440.0002
Average225.53225.52225.46225.47229.75229.67395.27

The bottom line shows the average highest score, rounded to the nearest 100th. While not particularly meaningful, it was curious to see the deal average (395.27) fall just under 400 — a reminder of the times we almost made three notrump.

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© 2017 Richard Pavlicek