Study 8J29 Main

Bag of Tricks

by Richard Pavlicek

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 Tricks Highest Contract Highest 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 DD Stats and General Stats.

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.

Study 8J29 Main Top Bag of Tricks

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.

Tricks North South West East N-S W-E Deal
13 1.4540 1.4532 1.4555 1.4557 1.4743 1.4763 2.9506
12 5.0672 5.0677 5.0618 5.0627 5.3151 5.3082 10.6160
11 10.3043 10.3029 10.3069 10.3046 10.4966 10.5025 20.7847
10 15.2467 15.2528 15.2429 15.2421 15.3821 15.3738 28.3820
9 17.8539 17.8496 17.8556 17.8559 17.8721 17.8762 24.9277
8 17.2525 17.2523 17.2498 17.2531 17.1567 17.1548 11.0335
7 14.0388 14.0374 14.0174 14.0178 13.8781 13.8536 1.3054
6 9.6342 9.6340 9.6580 9.6560 9.4763 9.5023 0.0002
5 5.5166 5.5181 5.5219 5.5203 5.4039 5.4084 0
4 2.5152 2.5148 2.5175 2.5198 2.4568 2.4599 0
3 0.8869 0.8877 0.8810 0.8804 0.8644 0.8580 0
2 0.2085 0.2083 0.2110 0.2110 0.2028 0.2055 0
1 0.0210 0.0210 0.0204 0.0205 0.0205 0.0200 0
0 0.0002 0.0002 0.0003 0.0003 0.0002 0.0003 0
Average 8.4105 8.4104 8.4100 8.4100 8.4420 8.4416 9.9996

Tricks	North	South	West	East	N-S	W-E	Deal
13	1.4540	1.4532	1.4555	1.4557	1.4743	1.4763	2.9506
12	5.0672	5.0677	5.0618	5.0627	5.3151	5.3082	10.6160
11	10.3043	10.3029	10.3069	10.3046	10.4966	10.5025	20.7847
10	15.2467	15.2528	15.2429	15.2421	15.3821	15.3738	28.3820
9	17.8539	17.8496	17.8556	17.8559	17.8721	17.8762	24.9277
8	17.2525	17.2523	17.2498	17.2531	17.1567	17.1548	11.0335
7	14.0388	14.0374	14.0174	14.0178	13.8781	13.8536	1.3054
6	9.6342	9.6340	9.6580	9.6560	9.4763	9.5023	0.0002
5	5.5166	5.5181	5.5219	5.5203	5.4039	5.4084	0
4	2.5152	2.5148	2.5175	2.5198	2.4568	2.4599	0
3	0.8869	0.8877	0.8810	0.8804	0.8644	0.8580	0
2	0.2085	0.2083	0.2110	0.2110	0.2028	0.2055	0
1	0.0210	0.0210	0.0204	0.0205	0.0205	0.0200	0
0	0.0002	0.0002	0.0003	0.0003	0.0002	0.0003	0
Average	8.4105	8.4104	8.4100	8.4100	8.4420	8.4416	9.9996

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.

Study 8J29 Main Top Bag of Tricks

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 ) indicated as “strain tricks”; e.g., 6 = hearts winning six tricks.

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

Contract North South West East N-S W-E Deal
7 NT 0.6896 0.6898 0.6875 0.6874 0.6952 0.6931 1.3883
7 0.2342 0.2342 0.2353 0.2356 0.2405 0.2416 0.4821
7 0.2040 0.2038 0.2048 0.2046 0.2084 0.2093 0.4177
7 0.1772 0.1760 0.1781 0.1777 0.1795 0.1808 0.3602
7 0.1491 0.1495 0.1498 0.1503 0.1508 0.1515 0.3023
6 NT 1.5467 1.5473 1.5378 1.5393 1.7237 1.7152 3.4389
6 1.1396 1.1401 1.1413 1.1406 1.1580 1.1589 2.3164
6 0.9576 0.9576 0.9586 0.9592 0.9760 0.9765 1.9508
6 0.7850 0.7859 0.7883 0.7881 0.8038 0.8056 1.6069
6 0.6382 0.6367 0.6357 0.6355 0.6537 0.6521 1.3029
5 NT 1.9900 1.9916 1.9895 1.9892 2.1120 2.1115 4.2232
5 2.6623 2.6616 2.6563 2.6541 2.6834 2.6781 5.3408
5 2.2466 2.2454 2.2586 2.2604 2.2652 2.2790 4.4911
5 1.8793 1.8785 1.8744 1.8726 1.8950 1.8892 3.7165
5 1.5262 1.5258 1.5281 1.5283 1.5410 1.5447 3.0132
4 NT 2.1616 2.1638 2.1620 2.1619 2.2599 2.2575 4.5097
4 4.1379 4.1347 4.1191 4.1206 4.1496 4.1320 7.9413
4 3.5127 3.5141 3.5269 3.5224 3.5221 3.5340 6.4593
4 2.9639 2.9674 2.9688 2.9710 2.9733 2.9788 5.2501
4 2.4706 2.4729 2.4661 2.4661 2.4772 2.4716 4.2216
3 NT 2.1694 2.1665 2.1684 2.1736 2.2295 2.2317 4.3646
3 4.9062 4.9039 4.8982 4.8925 4.8954 4.8870 7.6470
3 4.1908 4.1935 4.1985 4.2010 4.1826 4.1901 5.6063
3 3.5710 3.5688 3.5741 3.5724 3.5603 3.5629 4.1696
3 3.0165 3.0170 3.0163 3.0164 3.0044 3.0045 3.1401
2 NT 1.9152 1.9156 1.9190 1.9184 1.9406 1.9469 3.2578
2 4.8048 4.8038 4.7989 4.8022 4.7719 4.7686 3.7326
2 4.0912 4.0894 4.0981 4.1003 4.0577 4.0685 2.1254
2 3.4766 3.4782 3.4809 3.4782 3.4490 3.4479 1.2263
2 2.9647 2.9654 2.9529 2.9540 2.9374 2.9229 0.6913
1 NT 1.4422 1.4414 1.4440 1.4387 1.4472 1.4453 1.0566
1 4.0203 4.0220 4.0190 4.0193 3.9734 3.9719 0.2152
1 3.3700 3.3660 3.3590 3.3611 3.3266 3.3166 0.0283
1 2.8250 2.8276 2.8279 2.8319 2.7860 2.7901 0.0043
1 2.3812 2.3804 2.3676 2.3669 2.3450 2.3297 0.0008
NT 6 1.0274 1.0287 1.0305 1.0296 1.0221 1.0260 0.0002
6 2.8174 2.8166 2.8235 2.8242 2.7710 2.7796 0.0000
6 2.3173 2.3179 2.3226 2.3202 2.2762 2.2812 0.0000
6 1.9048 1.9045 1.9097 1.9091 1.8695 1.8742 0
6 1.5674 1.5662 1.5716 1.5729 1.5375 1.5413 0
NT 5 0.5989 0.5986 0.5943 0.5957 0.5906 0.5868 0
5 1.6657 1.6680 1.6729 1.6719 1.6332 1.6380 0
5 1.3390 1.3402 1.3374 1.3369 1.3112 1.3098 0
5 1.0653 1.0642 1.0667 1.0654 1.0406 1.0428 0
5 0.8476 0.8472 0.8506 0.8503 0.8282 0.8311 0
NT 4 0.2691 0.2695 0.2698 0.2708 0.2644 0.2650 0
4 0.7959 0.7966 0.8039 0.8042 0.7778 0.7860 0
4 0.6171 0.6160 0.6132 0.6128 0.6015 0.5981 0
4 0.4708 0.4708 0.4721 0.4727 0.4600 0.4610 0
4 0.3623 0.3618 0.3586 0.3593 0.3530 0.3498 0
NT 3 0.0957 0.0953 0.0939 0.0934 0.0929 0.0912 0
3 0.3026 0.3022 0.2989 0.2995 0.2949 0.2917 0
3 0.2202 0.2210 0.2188 0.2190 0.2151 0.2137 0
3 0.1572 0.1578 0.1573 0.1568 0.1531 0.1527 0
3 0.1112 0.1115 0.1122 0.1117 0.1084 0.1088 0
NT 2 0.0288 0.0288 0.0296 0.0294 0.0278 0.0288 0
2 0.0785 0.0786 0.0791 0.0793 0.0766 0.0771 0
2 0.0519 0.0516 0.0512 0.0510 0.0504 0.0498 0
2 0.0306 0.0307 0.0318 0.0319 0.0299 0.0312 0
2 0.0187 0.0185 0.0193 0.0193 0.0180 0.0188 0
NT 1 0.0060 0.0060 0.0056 0.0058 0.0058 0.0055 0
1 0.0084 0.0084 0.0084 0.0083 0.0082 0.0082 0
1 0.0041 0.0041 0.0041 0.0041 0.0040 0.0040 0
1 0.0020 0.0019 0.0017 0.0017 0.0019 0.0017 0
1 0.0006 0.0007 0.0006 0.0006 0.0006 0.0006 0
0 0.0002 0.0002 0.0003 0.0003 0.0002 0.0003 0
Average 2 NT 2 NT 2 NT 2 NT 2 NT 2 NT 4

Contract	North	South	West	East	N-S	W-E	Deal
7 NT	0.6896	0.6898	0.6875	0.6874	0.6952	0.6931	1.3883
7	0.2342	0.2342	0.2353	0.2356	0.2405	0.2416	0.4821
7	0.2040	0.2038	0.2048	0.2046	0.2084	0.2093	0.4177
7	0.1772	0.1760	0.1781	0.1777	0.1795	0.1808	0.3602
7	0.1491	0.1495	0.1498	0.1503	0.1508	0.1515	0.3023
6 NT	1.5467	1.5473	1.5378	1.5393	1.7237	1.7152	3.4389
6	1.1396	1.1401	1.1413	1.1406	1.1580	1.1589	2.3164
6	0.9576	0.9576	0.9586	0.9592	0.9760	0.9765	1.9508
6	0.7850	0.7859	0.7883	0.7881	0.8038	0.8056	1.6069
6	0.6382	0.6367	0.6357	0.6355	0.6537	0.6521	1.3029
5 NT	1.9900	1.9916	1.9895	1.9892	2.1120	2.1115	4.2232
5	2.6623	2.6616	2.6563	2.6541	2.6834	2.6781	5.3408
5	2.2466	2.2454	2.2586	2.2604	2.2652	2.2790	4.4911
5	1.8793	1.8785	1.8744	1.8726	1.8950	1.8892	3.7165
5	1.5262	1.5258	1.5281	1.5283	1.5410	1.5447	3.0132
4 NT	2.1616	2.1638	2.1620	2.1619	2.2599	2.2575	4.5097
4	4.1379	4.1347	4.1191	4.1206	4.1496	4.1320	7.9413
4	3.5127	3.5141	3.5269	3.5224	3.5221	3.5340	6.4593
4	2.9639	2.9674	2.9688	2.9710	2.9733	2.9788	5.2501
4	2.4706	2.4729	2.4661	2.4661	2.4772	2.4716	4.2216
3 NT	2.1694	2.1665	2.1684	2.1736	2.2295	2.2317	4.3646
3	4.9062	4.9039	4.8982	4.8925	4.8954	4.8870	7.6470
3	4.1908	4.1935	4.1985	4.2010	4.1826	4.1901	5.6063
3	3.5710	3.5688	3.5741	3.5724	3.5603	3.5629	4.1696
3	3.0165	3.0170	3.0163	3.0164	3.0044	3.0045	3.1401
2 NT	1.9152	1.9156	1.9190	1.9184	1.9406	1.9469	3.2578
2	4.8048	4.8038	4.7989	4.8022	4.7719	4.7686	3.7326
2	4.0912	4.0894	4.0981	4.1003	4.0577	4.0685	2.1254
2	3.4766	3.4782	3.4809	3.4782	3.4490	3.4479	1.2263
2	2.9647	2.9654	2.9529	2.9540	2.9374	2.9229	0.6913
1 NT	1.4422	1.4414	1.4440	1.4387	1.4472	1.4453	1.0566
1	4.0203	4.0220	4.0190	4.0193	3.9734	3.9719	0.2152
1	3.3700	3.3660	3.3590	3.3611	3.3266	3.3166	0.0283
1	2.8250	2.8276	2.8279	2.8319	2.7860	2.7901	0.0043
1	2.3812	2.3804	2.3676	2.3669	2.3450	2.3297	0.0008
NT 6	1.0274	1.0287	1.0305	1.0296	1.0221	1.0260	0.0002
6	2.8174	2.8166	2.8235	2.8242	2.7710	2.7796	0.0000
6	2.3173	2.3179	2.3226	2.3202	2.2762	2.2812	0.0000
6	1.9048	1.9045	1.9097	1.9091	1.8695	1.8742	0
6	1.5674	1.5662	1.5716	1.5729	1.5375	1.5413	0
NT 5	0.5989	0.5986	0.5943	0.5957	0.5906	0.5868	0
5	1.6657	1.6680	1.6729	1.6719	1.6332	1.6380	0
5	1.3390	1.3402	1.3374	1.3369	1.3112	1.3098	0
5	1.0653	1.0642	1.0667	1.0654	1.0406	1.0428	0
5	0.8476	0.8472	0.8506	0.8503	0.8282	0.8311	0
NT 4	0.2691	0.2695	0.2698	0.2708	0.2644	0.2650	0
4	0.7959	0.7966	0.8039	0.8042	0.7778	0.7860	0
4	0.6171	0.6160	0.6132	0.6128	0.6015	0.5981	0
4	0.4708	0.4708	0.4721	0.4727	0.4600	0.4610	0
4	0.3623	0.3618	0.3586	0.3593	0.3530	0.3498	0
NT 3	0.0957	0.0953	0.0939	0.0934	0.0929	0.0912	0
3	0.3026	0.3022	0.2989	0.2995	0.2949	0.2917	0
3	0.2202	0.2210	0.2188	0.2190	0.2151	0.2137	0
3	0.1572	0.1578	0.1573	0.1568	0.1531	0.1527	0
3	0.1112	0.1115	0.1122	0.1117	0.1084	0.1088	0
NT 2	0.0288	0.0288	0.0296	0.0294	0.0278	0.0288	0
2	0.0785	0.0786	0.0791	0.0793	0.0766	0.0771	0
2	0.0519	0.0516	0.0512	0.0510	0.0504	0.0498	0
2	0.0306	0.0307	0.0318	0.0319	0.0299	0.0312	0
2	0.0187	0.0185	0.0193	0.0193	0.0180	0.0188	0
NT 1	0.0060	0.0060	0.0056	0.0058	0.0058	0.0055	0
1	0.0084	0.0084	0.0084	0.0083	0.0082	0.0082	0
1	0.0041	0.0041	0.0041	0.0041	0.0040	0.0040	0
1	0.0020	0.0019	0.0017	0.0017	0.0019	0.0017	0
1	0.0006	0.0007	0.0006	0.0006	0.0006	0.0006	0
0	0.0002	0.0002	0.0003	0.0003	0.0002	0.0003	0
Average	2 NT	2 NT	2 NT	2 NT	2 NT	2 NT	4

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.

Study 8J29 Main Top Bag of Tricks

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.

Score North South West East N-S W-E Deal
1520 0.6896 0.6898 0.6875 0.6874 0.6952 0.6931 1.3883
1510 0.4382 0.4379 0.4401 0.4402 0.4489 0.4509 0.8998
1440 0.3263 0.3256 0.3279 0.3281 0.3302 0.3323 0.6625
990 1.5467 1.5473 1.5378 1.5393 1.7237 1.7152 3.4389
980 2.0972 2.0977 2.0999 2.0998 2.1340 2.1354 4.2672
920 1.4232 1.4227 1.4240 1.4236 1.4574 1.4576 2.9098
460 1.9900 1.9916 1.9895 1.9892 2.1120 2.1115 4.2232
450 4.9089 4.9070 4.9149 4.9145 4.9486 4.9571 9.8318
430 3.0409 3.0418 3.0378 3.0363 3.1800 3.1738 6.3476
420 8.4078 8.4077 8.4047 8.4021 8.4346 8.4287 16.1091
400 5.1762 5.1739 5.1693 5.1719 5.2573 5.2528 10.0894
140 10.3611 10.3614 10.3569 10.3569 10.3433 10.3404 16.5232
130 2.9325 2.9360 2.9420 2.9429 2.9105 2.9208 3.6603
120 3.3365 3.3363 3.3379 3.3385 3.3767 3.3813 5.9305
110 14.0622 14.0583 14.0685 14.0712 13.9583 13.9701 10.4950
90 7.8835 7.8849 7.8779 7.8708 7.8337 7.8160 2.9743
80 7.3904 7.3880 7.3780 7.3803 7.3000 7.2885 0.2435
70 5.2062 5.2080 5.1955 5.1988 5.1309 5.1198 0.0052
0 18.7827 18.7842 18.8101 18.8082 18.4250 18.4544 0.0002
Average 225.53 225.52 225.46 225.47 229.75 229.67 395.27

Score	North	South	West	East	N-S	W-E	Deal
1520	0.6896	0.6898	0.6875	0.6874	0.6952	0.6931	1.3883
1510	0.4382	0.4379	0.4401	0.4402	0.4489	0.4509	0.8998
1440	0.3263	0.3256	0.3279	0.3281	0.3302	0.3323	0.6625
990	1.5467	1.5473	1.5378	1.5393	1.7237	1.7152	3.4389
980	2.0972	2.0977	2.0999	2.0998	2.1340	2.1354	4.2672
920	1.4232	1.4227	1.4240	1.4236	1.4574	1.4576	2.9098
460	1.9900	1.9916	1.9895	1.9892	2.1120	2.1115	4.2232
450	4.9089	4.9070	4.9149	4.9145	4.9486	4.9571	9.8318
430	3.0409	3.0418	3.0378	3.0363	3.1800	3.1738	6.3476
420	8.4078	8.4077	8.4047	8.4021	8.4346	8.4287	16.1091
400	5.1762	5.1739	5.1693	5.1719	5.2573	5.2528	10.0894
140	10.3611	10.3614	10.3569	10.3569	10.3433	10.3404	16.5232
130	2.9325	2.9360	2.9420	2.9429	2.9105	2.9208	3.6603
120	3.3365	3.3363	3.3379	3.3385	3.3767	3.3813	5.9305
110	14.0622	14.0583	14.0685	14.0712	13.9583	13.9701	10.4950
90	7.8835	7.8849	7.8779	7.8708	7.8337	7.8160	2.9743
80	7.3904	7.3880	7.3780	7.3803	7.3000	7.2885	0.2435
70	5.2062	5.2080	5.1955	5.1988	5.1309	5.1198	0.0052
0	18.7827	18.7842	18.8101	18.8082	18.4250	18.4544	0.0002
Average	225.53	225.52	225.46	225.47	229.75	229.67	395.27

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 all the times we almost made three notrump.

Study 8J29 Main Top Bag of Tricks