Study 8J17   Main

Hand Evaluation Stats

by Richard Pavlicek

Virtually all bridge players are familiar with the measurements of a bridge hand, but few are aware of the frequency of occurrence or the percentages involved. For instance, what is the chance of holding at least 13 HCP? Or at most 5 losers? Or exactly 4 controls? Or a balanced hand? Or a 6-card suit? Answers to these and a myriad of other questions can be found here.

HCP

Controls

Guards

Playing Tricks

Losing Tricks

Defensive Tricks

Freakness

Longest

Shortest

All statistics were found by counting the number of bridge hands (out of 635,013,559,600 possible) that fit each listed quantity for each measurement, in some cases an extremely complicated task. Numbers are shown to four decimal places unless exact, so 0.0000 is greater than zero, and 100.0000 (percent) is less than 100, but simply round that way to the nearest 10,000th.

High-Card Points

Every bridge player knows about high-card points (HCP), invented by Milton Work about a century ago, before Contract Bridge even existed. In case you’ve been living in a cave: Each ace = 4, each king = 3, each queen = 2, and each jack = 1.

High-Card Points Summary
Hands evaluated	635013559600
Unique values	38
Minimum	0
Maximum	37
Mode	10
Median	10
Mean	10
Standard deviation	4.1302

The following table shows the number of hands and percent chance for each number of HCP. Percents are shown in three ways: for the specific number, at least that number, and at most that number. To find the chance of an interior range (e.g., 15-17 HCP) add the specific percents of each number in that range.

HCP	Hands	Specific	At Least	At Most
0	2310789600	0.3639	100	0.3639
1	5006710800	0.7884	99.6361	1.1523
2	8611542576	1.3561	98.8477	2.5085
3	15636342960	2.4624	97.4915	4.9708
4	24419055136	3.8454	95.0292	8.8163
5	32933031040	5.1862	91.1837	14.0025
6	41619399184	6.5541	85.9975	20.5565
7	50979441968	8.0281	79.4435	28.5846
8	56466608128	8.8922	71.4154	37.4768
9	59413313872	9.3562	62.5232	46.8331
10	59723754816	9.4051	53.1669	56.2382
11	56799933520	8.9447	43.7618	65.1828
12	50971682080	8.0269	34.8172	73.2097
13	43906944752	6.9143	26.7903	80.1240
14	36153374224	5.6933	19.8760	85.8174
15	28090962724	4.4237	14.1826	90.2410
16	21024781756	3.3109	9.7590	93.5520
17	14997082848	2.3617	6.4480	95.9137
18	10192504020	1.6051	4.0863	97.5187
19	6579838440	1.0362	2.4813	98.5549
20	4086538404	0.6435	1.4451	99.1985
21	2399507844	0.3779	0.8015	99.5763
22	1333800036	0.2100	0.4237	99.7864
23	710603628	0.1119	0.2136	99.8983
24	354993864	0.0559	0.1017	99.9542
25	167819892	0.0264	0.0458	99.9806
26	74095248	0.0117	0.0194	99.9923
27	31157940	0.0049	0.0077	99.9972
28	11790760	0.0019	0.0028	99.9990
29	4236588	0.0007	0.0010	99.9997
30	1396068	0.0002	0.0003	99.9999
31	388196	0.0001	0.0001	100.0000
32	109156	0.0000	0.0000	100.0000
33	22360	0.0000	0.0000	100.0000
34	4484	0.0000	0.0000	100.0000
35	624	0.0000	0.0000	100.0000
36	60	0.0000	0.0000	100.0000
37	4	0.0000	0.0000	100
U = 38	635013559600	100	3900

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Controls

Controls are counted simply as 2 for each ace and 1 for each king.

Controls Summary
Hands evaluated	635013559600
Unique values	13
Minimum	0
Maximum	12
Mode	3
Median	3
Mean	3
Standard deviation	1.8150

The following table shows the number of hands and percent chance of each number of controls. Percents are shown in three ways: for the specific number, at least that number, and at most that number. To find the chance of an interior range (e.g., 3-4 controls) add the specific percents of each number in that range.

Controls	Hands	Specific	At Least	At Most
0	51915526432	8.1755	100	8.1755
1	84362730452	13.2852	91.8245	21.4607
2	130378765244	20.5317	78.5393	41.9923
3	132634453224	20.8869	58.0077	62.8792
4	106275127972	16.7359	37.1208	79.6151
5	70893050800	11.1640	20.3849	90.7791
6	36155455908	5.6937	9.2209	96.4728
7	15596471176	2.4561	3.5272	98.9288
8	5192436964	0.8177	1.0712	99.7465
9	1322059596	0.2082	0.2535	99.9547
10	258159616	0.0407	0.0453	99.9954
11	28236208	0.0044	0.0046	99.9998
12	1086008	0.0002	0.0002	100
U = 13	635013559600	100	1400

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Guards

Guards is a measurement I invented to define protection for notrump play. A suit is guarded (stopped) if at least as good as A, K-x, Q-x-x, J-x-x-x, 10-x-x-x-x or any six cards; partially guarded (half a stopper) if at least as good as K, Q-x, J-x-x, 10-x-x-x or any five cards; and twice guarded (double stopper) if at least as good as A-Q, K-J-x, Q-10-x-x or J-9-x-x-x. Guards assume you are behind the missing honors, or the suit is led around to you, so might not fulfill in practice, but experience has shown that optimism is a winning trait.

While a suit could also be thrice guarded (triple stopper) or more, this would have little if any relevance in actual play, so a double guard includes anything stronger as well. Therefore, each suit can have zero, 0.5, 1 or 2 guards, and the range for a bridge hand is 0 to 8 in 0.5 increments, except 7.5 is impossible.

Guards Summary
Hands evaluated	635013559600
Unique values	16
Minimum	0
Maximum	8
Mode	4
Median	4
Mean	3.7757
Standard deviation	TBD

The following table shows the number of hands and percent chance of each number of guards. Percents are shown in three ways: for the specific number, at least that number, and at most that number. To find the chance of an interior range (e.g., 4-5 guards) add the specific percents of each number in that range.

Guards	Hands	Specific	At Least	At Most
0	403313120	0.0635	100	0.0635
0.5	1282854496	0.2020	99.9365	0.2655
1	8347621540	1.3146	99.7345	1.5801
1.5	11620495420	1.8300	98.4199	3.4101
2	48343143440	7.6129	96.5899	11.0230
2.5	42415369052	6.6794	88.9770	17.7024
3	118469367036	18.6562	82.2976	36.3586
3.5	60262891660	9.4900	63.6414	45.8486
4	148255312584	23.3468	54.1514	69.1954
4.5	46267945572	7.2861	30.8046	76.4816
5	94882534192	14.9418	23.5184	91.4234
5.5	14174326212	2.2321	8.5766	93.6555
6	32675670276	5.1457	6.3445	98.8012
6.5	2291913396	0.3609	1.1988	99.1621
7	4960141408	0.7811	0.8379	99.9432
8	360660196	0.0568	0.0568	100
U = 16	635013559600	100	1700

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Playing Tricks

Playing tricks is a measurement of trick-taking potential with your longest suit trumps. This is typically used to decide how high to bid with a 6+ card suit (more often 7+) but can be applied any time. The top three cards in each suit are estimated for trick production (see chart) and 1 trick is added for each card over three in any suit; hence every hand has at least 1 playing trick. Common practice is to consider half-trick increments, implying that a full trick will materialize about half the time.

K, Q-x, K-x, J-10-x, Q-x-x	0.5
A, K-J, K-Q, A-x, Q-J-x, K-x-x, A-x-x	1
A-J, A-Q, K-J-10, K-Q-x, A-J-x, A-Q-x	1.5
A-K, K-Q-J, A-Q-10, A-K-x	2
A-Q-J, A-K-J	2.5
A-K-Q	3

Refinements: In an 8-card suit each listed holding with an ‘x’ (except Q-J-x) is increased by half a trick, e.g., A-x-x-x-x-x-x-x = 6.5 tricks. In a 9 or 10-card suit, only the top two cards matter (each card over two is a trick), e.g., A-K-x-x-x-x-x-x-x = 9 tricks. And for the real dreamers, with 11+ cards only the ace matters; if you have it assume all winners, else all but one.

Playing Tricks Summary
Hands evaluated	635013559600
Unique values	25
Minimum	1
Maximum	13
Mode	5
Median	5.5
Mean	5.3685
Standard deviation	1.5783

The following table shows the number of hands and percent chance for each number of playing tricks. Percents are shown in three ways: for the specific number, at least that number, and at most that number. To find the chance of an interior range (e.g., 6.5 to 7.5 playing tricks) add the specific percents of each number in that range.

Tricks	Hands	Specific	At Least	At Most
1	1022787584	0.1611	100	0.1611
1.5	1834835968	0.2889	99.8389	0.4500
2	8288385280	1.3052	99.5500	1.7552
2.5	13587961408	2.1398	98.2448	3.8950
3	29907404912	4.7097	96.1050	8.6048
3.5	41453445520	6.5280	91.3952	15.1327
4	61513033480	9.6869	84.8673	24.8196
4.5	70986532496	11.1787	75.1804	35.9983
5	80169517920	12.6249	64.0017	48.6232
5.5	77238005980	12.1632	51.3768	60.7864
6	70723648784	11.1373	39.2136	71.9237
6.5	57473116632	9.0507	28.0763	80.9744
7	44455847496	7.0008	19.0256	87.9752
7.5	30812461108	4.8523	12.0248	92.8275
8	20488020544	3.2264	7.1725	96.0539
8.5	12080979332	1.9025	3.9461	97.9563
9	6865130592	1.0811	2.0437	99.0374
9.5	3412850548	0.5374	0.9626	99.5749
10	1635279380	0.2575	0.4251	99.8324
10.5	673249156	0.1060	0.1676	99.9384
11	269155616	0.0424	0.0616	99.9808
11.5	86893416	0.0137	0.0192	99.9945
12	27926304	0.0044	0.0055	99.9989
12.5	5652256	0.0009	0.0011	99.9998
13	1437888	0.0002	0.0002	100
U = 25	635013559600	100	2600

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Losing Tricks

Losing trick count (LTC) is a method of evaluating hands for suit play. Each suit has 0-3 losers, so an entire bridge hand has 0-12 losers. More specifically, a void suit = 0 losers; a singleton = 1 loser (unless the ace, then 0); a doubleton = 2 losers (unless A-K, then 0, or with ace or king = 1); suits of 3+ cards have one loser for each missing ace, king and queen.

Refinements: In a suit of 8-10 cards only the ace and king matter; i.e., A-K = 0 losers, A or K = 1 loser, else 2 losers. (With exactly 8 cards this is slightly optimistic — a half loser would be fairer for a missing queen — but LTC advocates rarely use fractions, and going long is better than no adjustment at all.) With 11+ cards only the ace matters, hence 0 or 1 loser accordingly.

Losing Tricks Summary
Hands evaluated	635013559600
Unique values	13
Minimum	0
Maximum	12
Mode	8
Median	8
Mean	7.5566
Standard deviation	1.5739

The following table shows the number of hands and percent chance for each losing trick count. Percents are shown in three ways: for the specific number, at least that number, and at most that number. To find the chance of an interior range (e.g., 4-5 losers) add the specific percents of each number in that range.

Losers	Hands	Specific	At Least	At Most
0	1611768	0.0003	100	0.0003
1	39329036	0.0062	99.9997	0.0064
2	454375244	0.0716	99.9936	0.0780
3	3126502788	0.4924	99.9220	0.5704
4	14069567832	2.2156	99.4296	2.7860
5	43541225304	6.8567	97.2140	9.6427
6	94901190408	14.9448	90.3573	24.5875
7	145807470468	22.9613	75.4125	47.5488
8	155515164912	24.4901	52.4512	72.0388
9	111719209440	17.5932	27.9612	89.6320
10	51111464400	8.0489	10.3680	97.6809
11	13274928000	2.0905	2.3191	99.7714
12	1451520000	0.2286	0.2286	100
U = 13	635013559600	100	1400

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Defensive Tricks

Defensive tricks, also known as “quick tricks,” is a measurement of defensive prospects with your shortest suit trumps. At most 2 tricks can be counted in each suit per the formula: A-K = 2, A-Q = 1.5, A or K-Q = 1, K = 0.5; hence the range for a bridge hand is 0 to 8. Note that a king does not have to be guarded, as a singleton king usually has the same chance to win a trick.

Refinements: In a 7 or 8-card suit only one defensive trick can be counted (no halves) for which you must have the ace. For example, A-K-x-x-x-x-x = 1 (only a fool would expect A-K to cash) and K-Q-J-x-x-x-x = 0. Further, you cannot count any defensive trick in a 9+ card suit — not a likely issue since you’d rarely be defending.

Flukes: A hand with A-K-Q in every suit arguably has 9 defensive tricks, since a third trick is assured in whichever suit is trumps; similarly, at least A-K-J in every suit is arguably worth 8.5 tricks. These rare cases are ignored, so the maximum per suit is always 2.

Defensive Tricks Summary
Hands evaluated	635013559600
Unique values	17
Minimum	0
Maximum	8
Mode	1
Median	1.5
Mean	1.7782
Standard deviation	1.1248

The following table shows the number of hands and percent chance for each number of defensive tricks. Percents are shown in three ways: for the specific number, at least that number, and at most that number. To find the chance of an interior range (e.g., 2.5 to 3 defensive tricks) add the specific percents of each number in that range.

Tricks	Hands	Specific	At Least	At Most
0	54389790360	8.5651	100	8.5651
0.5	61602628572	9.7010	91.4349	18.2661
1	113660108928	17.8988	81.7339	36.1650
1.5	96416902372	15.1834	63.8350	51.3484
2	111050090784	17.4878	48.6516	68.8362
2.5	72935403696	11.4856	31.1638	80.3219
3	59887556392	9.4309	19.6781	89.7528
3.5	31112636760	4.8995	10.2472	94.6523
4	20067427740	3.1602	5.3477	97.8125
4.5	7980162216	1.2567	2.1875	99.0692
5	4055992444	0.6387	0.9308	99.7079
5.5	1214000280	0.1912	0.2921	99.8991
6	502896536	0.0792	0.1009	99.9783
6.5	99889520	0.0157	0.0217	99.9940
7	33145000	0.0052	0.0060	99.9992
7.5	3843840	0.0006	0.0008	99.9998
8	1084160	0.0002	0.0002	100
U = 17	635013559600	100	1800

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Freakness

Many years ago I devised a method to rank the 39 generic hand patterns on a linear scale. The formula counts 1 point for each card over four or under three in each suit, plus 1 extra point if the hand has any singleton (or 2 extra points if the hand has any void). This creates a simple scale that I have found useful in many ways. For instance, a balanced hand is easily defined as having freakness less than 3.

4-3-3-3	0
4-4-3-2	1
5-3-3-2	2
4-4-4-1, 5-4-2-2	3
5-4-3-1, 6-3-2-2	4
6-3-3-1	5
5-4-4-0, 5-5-2-1, 6-4-2-1, 7-2-2-2	6
5-5-3-0, 6-4-3-0, 7-3-2-1	7
6-5-1-1, 7-3-3-0, 7-4-1-1	8
6-5-2-0, 7-4-2-0, 8-2-2-1, 8-3-1-1	9
8-3-2-0	10
6-6-1-0, 7-5-1-0, 8-4-1-0, 9-2-1-1	11
9-2-2-0, 9-3-1-0	12
7-6-0-0, 8-5-0-0, 9-4-0-0, 10-1-1-1	13
10-2-1-0, 10-3-0-0	14
11-1-1-0	16
11-2-0-0	16
12-1-0-0	18
13-0-0-0	20

Freakness Summary
Hands evaluated	635013559600
Unique values	18
Minimum	0
Maximum	20
Mode	1
Median	3
Mean	2.9829
Standard deviation	2.2056

The following table shows the number of hands and percent chance for each freakness (missing numbers 15, 17 and 19 are impossible by the formula). Percents are shown in three ways: for the specific number, at least that number, and at most that number. To find the chance of an interior range (e.g., freakness 3-4) add the specific percents of each number in that range.

Freakness	Hands	Specific	At Least	At Most
0	66905856160	10.5361	100	10.5361
1	136852887600	21.5512	89.4639	32.0873
2	98534079072	15.5168	67.9127	47.6042
3	86189672140	13.5729	52.3958	61.1770
4	117942306768	18.5732	38.8230	79.7502
5	21896462016	3.4482	20.2498	83.1984
6	61166193660	9.6323	16.8016	92.8307
7	26049899304	4.1023	7.1693	96.9329
8	8651399328	1.3624	3.0671	98.2953
9	8399095848	1.3227	1.7047	99.6180
10	689049504	0.1085	0.3820	99.7265
11	1548621360	0.2439	0.2735	99.9704
12	116001600	0.0183	0.0296	99.9887
13	63860368	0.0101	0.0113	99.9987
14	7941648	0.0013	0.0013	100.0000
16	231192	0.0000	0.0000	100.0000
18	2028	0.0000	0.0000	100.0000
20	4	0.0000	0.0000	100
U = 18	635013559600	100	1900

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Longest Suit

The longest suit of a bridge hand must be from 4 to 13 cards. This was once challenged by my Uncle Cedric who opened 1  on Q-x-x in a drunken stupor. When asked about the psych he blathered, “It wasn’t a friggin’ psych! What else could I bid with five doubletons?”

Longest Suit Summary
Hands evaluated	635013559600
Unique values	10
Minimum	4
Maximum	13
Mode	5
Median	5
Mean	4.9008
Standard deviation	0.8342

The following table shows the number of hands and percent chance for each longest suit. Percents are shown in three ways: for the specific number, at least that number, and at most that number. To find the chance of an interior range (e.g., 6-7 card suit) add the specific percents of each number in that range.

Longest Suit	Hands	Specific	At Least	At Most
4	222766089260	35.0805	100	35.0805
5	281562853572	44.3397	64.9195	79.4202
6	105080049360	16.5477	20.5798	95.9679
7	22394644272	3.5266	4.0321	99.4945
8	2963997036	0.4668	0.5055	99.9613
9	235237860	0.0370	0.0387	99.9983
10	10455016	0.0016	0.0017	100.0000
11	231192	0.0000	0.0000	100.0000
12	2028	0.0000	0.0000	100.0000
13	4	0.0000	0.0000	100
U = 10	635013559600	100	1100

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Shortest Suit

The shortest suit of a bridge hand must be from 0 to 3 cards.

Shortest Suit Summary
Hands evaluated	635013559600
Unique values	4
Minimum	0
Maximum	3
Mode	2
Median	2
Mean	1.6977
Standard deviation	0.7237

The following table shows the number of hands and percent chance for each shortest suit. Percents are shown in three ways: for the specific number, at least that number, and at most that number.

Shortest Suit	Hands	Specific	At Least	At Most
0	32427298180	5.1066	100	5.1066
1	194023212812	30.5542	94.8934	35.6607
2	341657192448	53.8031	64.3393	89.4639
3	66905856160	10.5361	10.5361	100
U = 4	635013559600	100	500

Check total of At Least + At Most should be 100(U+1). For example, if the specific percents for “Shortest Suit” are designated [abcd] then At Least column is a+b+c+d+b+c+d+c+d+d, and At Most column is a+a+b+a+b+c+a+b+c+d, so the sum of both columns is 5(a+b+c+d) or 500 percent.

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© 2012 Richard Pavlicek