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.

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 evaluated635013559600
Unique values38
Minimum0
Maximum37
Mode10
Median10
Mean10
Standard deviation4.1302

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.

HCPHandsSpecificAt LeastAt Most
023107896000.36391000.3639
150067108000.788499.63611.1523
286115425761.356198.84772.5085
3156363429602.462497.49154.9708
4244190551363.845495.02928.8163
5329330310405.186291.183714.0025
6416193991846.554185.997520.5565
7509794419688.028179.443528.5846
8564666081288.892271.415437.4768
9594133138729.356262.523246.8331
10597237548169.405153.166956.2382
11567999335208.944743.761865.1828
12509716820808.026934.817273.2097
13439069447526.914326.790380.1240
14361533742245.693319.876085.8174
15280909627244.423714.182690.2410
16210247817563.31099.759093.5520
17149970828482.36176.448095.9137
18101925040201.60514.086397.5187
1965798384401.03622.481398.5549
2040865384040.64351.445199.1985
2123995078440.37790.801599.5763
2213338000360.21000.423799.7864
237106036280.11190.213699.8983
243549938640.05590.101799.9542
251678198920.02640.045899.9806
26740952480.01170.019499.9923
27311579400.00490.007799.9972
28117907600.00190.002899.9990
2942365880.00070.001099.9997
3013960680.00020.000399.9999
313881960.00010.0001100.0000
321091560.00000.0000100.0000
33223600.00000.0000100.0000
3444840.00000.0000100.0000
356240.00000.0000100.0000
36600.00000.0000100.0000
3740.00000.0000100
U = 386350135596001003900

 Study 8J17   Main Top   Hand Evaluation Stats

## Controls

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

Controls Summary
Hands evaluated635013559600
Unique values13
Minimum0
Maximum12
Mode3
Median3
Mean3
Standard deviation1.8150

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.

ControlsHandsSpecificAt LeastAt Most
0519155264328.17551008.1755
18436273045213.285291.824521.4607
213037876524420.531778.539341.9923
313263445322420.886958.007762.8792
410627512797216.735937.120879.6151
57089305080011.164020.384990.7791
6361554559085.69379.220996.4728
7155964711762.45613.527298.9288
851924369640.81771.071299.7465
913220595960.20820.253599.9547
102581596160.04070.045399.9954
11282362080.00440.004699.9998
1210860080.00020.0002100
U = 136350135596001001400

 Study 8J17   Main Top   Hand Evaluation Stats

## 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 evaluated635013559600
Unique values16
Minimum0
Maximum8
Mode4
Median4
Mean3.7757
Standard deviationTBD

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.

GuardsHandsSpecificAt LeastAt Most
04033131200.06351000.0635
0.512828544960.202099.93650.2655
183476215401.314699.73451.5801
1.5116204954201.830098.41993.4101
2483431434407.612996.589911.0230
2.5424153690526.679488.977017.7024
311846936703618.656282.297636.3586
3.5602628916609.490063.641445.8486
414825531258423.346854.151469.1954
4.5462679455727.286130.804676.4816
59488253419214.941823.518491.4234
5.5141743262122.23218.576693.6555
6326756702765.14576.344598.8012
6.522919133960.36091.198899.1621
749601414080.78110.837999.9432
83606601960.05680.0568100
U = 166350135596001001700

 Study 8J17   Main Top   Hand Evaluation Stats

## 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 evaluated635013559600
Unique values25
Minimum1
Maximum13
Mode5
Median5.5
Mean5.3685
Standard deviation1.5783

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.

TricksHandsSpecificAt LeastAt Most
110227875840.16111000.1611
1.518348359680.288999.83890.4500
282883852801.305299.55001.7552
2.5135879614082.139898.24483.8950
3299074049124.709796.10508.6048
3.5414534455206.528091.395215.1327
4615130334809.686984.867324.8196
4.57098653249611.178775.180435.9983
58016951792012.624964.001748.6232
5.57723800598012.163251.376860.7864
67072364878411.137339.213671.9237
6.5574731166329.050728.076380.9744
7444558474967.000819.025687.9752
7.5308124611084.852312.024892.8275
8204880205443.22647.172596.0539
8.5120809793321.90253.946197.9563
968651305921.08112.043799.0374
9.534128505480.53740.962699.5749
1016352793800.25750.425199.8324
10.56732491560.10600.167699.9384
112691556160.04240.061699.9808
11.5868934160.01370.019299.9945
12279263040.00440.005599.9989
12.556522560.00090.001199.9998
1314378880.00020.0002100
U = 256350135596001002600

 Study 8J17   Main Top   Hand Evaluation Stats

## 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 evaluated635013559600
Unique values13
Minimum0
Maximum12
Mode8
Median8
Mean7.5566
Standard deviation1.5739

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.

LosersHandsSpecificAt LeastAt Most
016117680.00031000.0003
1393290360.006299.99970.0064
24543752440.071699.99360.0780
331265027880.492499.92200.5704
4140695678322.215699.42962.7860
5435412253046.856797.21409.6427
69490119040814.944890.357324.5875
714580747046822.961375.412547.5488
815551516491224.490152.451272.0388
911171920944017.593227.961289.6320
10511114644008.048910.368097.6809
11132749280002.09052.319199.7714
1214515200000.22860.2286100
U = 136350135596001001400

 Study 8J17   Main Top   Hand Evaluation Stats

## 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 evaluated635013559600
Unique values17
Minimum0
Maximum8
Mode1
Median1.5
Mean1.7782
Standard deviation1.1248

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.

TricksHandsSpecificAt LeastAt Most
0543897903608.56511008.5651
0.5616026285729.701091.434918.2661
111366010892817.898881.733936.1650
1.59641690237215.183463.835051.3484
211105009078417.487848.651668.8362
2.57293540369611.485631.163880.3219
3598875563929.430919.678189.7528
3.5311126367604.899510.247294.6523
4200674277403.16025.347797.8125
4.579801622161.25672.187599.0692
540559924440.63870.930899.7079
5.512140002800.19120.292199.8991
65028965360.07920.100999.9783
6.5998895200.01570.021799.9940
7331450000.00520.006099.9992
7.538438400.00060.000899.9998
810841600.00020.0002100
U = 176350135596001001800

 Study 8J17   Main Top   Hand Evaluation Stats

## 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 evaluated635013559600
Unique values18
Minimum0
Maximum20
Mode1
Median3
Mean2.9829
Standard deviation2.2056

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.

FreaknessHandsSpecificAt LeastAt Most
06690585616010.536110010.5361
113685288760021.551289.463932.0873
29853407907215.516867.912747.6042
38618967214013.572952.395861.1770
411794230676818.573238.823079.7502
5218964620163.448220.249883.1984
6611661936609.632316.801692.8307
7260498993044.10237.169396.9329
886513993281.36243.067198.2953
983990958481.32271.704799.6180
106890495040.10850.382099.7265
1115486213600.24390.273599.9704
121160016000.01830.029699.9887
13638603680.01010.011399.9987
1479416480.00130.0013100.0000
162311920.00000.0000100.0000
1820280.00000.0000100.0000
2040.00000.0000100
U = 186350135596001001900

 Study 8J17   Main Top   Hand Evaluation Stats

## 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 evaluated635013559600
Unique values10
Minimum4
Maximum13
Mode5
Median5
Mean4.9008
Standard deviation0.8342

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 SuitHandsSpecificAt LeastAt Most
422276608926035.080510035.0805
528156285357244.339764.919579.4202
610508004936016.547720.579895.9679
7223946442723.52664.032199.4945
829639970360.46680.505599.9613
92352378600.03700.038799.9983
10104550160.00160.0017100.0000
112311920.00000.0000100.0000
1220280.00000.0000100.0000
1340.00000.0000100
U = 106350135596001001100

 Study 8J17   Main Top   Hand Evaluation Stats

## Shortest Suit

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

Shortest Suit Summary
Hands evaluated635013559600
Unique values4
Minimum0
Maximum3
Mode2
Median2
Mean1.6977
Standard deviation0.7237

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 SuitHandsSpecificAt LeastAt Most
0324272981805.10661005.1066
119402321281230.554294.893435.6607
234165719244853.803164.339389.4639
36690585616010.536110.5361100
U = 4635013559600100500

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.

 Study 8J17   Main Top   Hand Evaluation Stats