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Study 8J17 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 | Points | Playing Tricks | Losing Tricks | Quick Tricks | Controls | Guards | 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.

*To evaluate a specific hand*

Bridge Hand Evaluator

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 |

In any point-count system, HCP must be augmented by distributional factors to obtain a fair appraisal of a bridge hand. To this endeavor, many reasonable methods exist. This study uses the Pavlicek Point Count (I can’t imagine why) which adds 3 points for a void, 2 points for a singleton, and 1 point for a doubleton — with an accurate adjustment for flawed (unprotected) honors. Further, an extra point is added for any four aces and/or 10s.

Revalued Points Summary | |
---|---|

Hands evaluated | 635013559600 |

Unique values | 39 |

Minimum | 0 |

Maximum | 38 |

Mode | 11 |

Median | 11 |

Mean | 11.3682 |

Standard deviation | 4.2781 |

The following table shows the number of hands and percent chance for each number of revalued points. 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., 13-15 points) add the specific percents of each number in that range.

Points | Hands | Specific | At Least | At Most |
---|---|---|---|---|

0 | 293805568 | 0.0463 | 100 | 0.0463 |

1 | 1766279872 | 0.2781 | 99.9537 | 0.3244 |

2 | 4460756944 | 0.7025 | 99.6756 | 1.0269 |

3 | 8243807120 | 1.2982 | 98.9731 | 2.3251 |

4 | 13933491492 | 2.1942 | 97.6749 | 4.5193 |

5 | 21764214176 | 3.4274 | 95.4807 | 7.9467 |

6 | 30423163944 | 4.7909 | 92.0533 | 12.7376 |

7 | 38944701748 | 6.1329 | 87.2624 | 18.8705 |

8 | 47115157372 | 7.4196 | 81.1295 | 26.2900 |

9 | 53631235236 | 8.4457 | 73.7100 | 34.7357 |

10 | 57295091856 | 9.0227 | 65.2643 | 43.7584 |

11 | 58143902808 | 9.1563 | 56.2416 | 52.9147 |

12 | 56245617636 | 8.8574 | 47.0853 | 61.7721 |

13 | 51805296432 | 8.1581 | 38.2279 | 69.9302 |

14 | 45528936116 | 7.1698 | 30.0698 | 77.1000 |

15 | 38297308552 | 6.0309 | 22.9000 | 83.1309 |

16 | 30847863460 | 4.8578 | 16.8691 | 87.9888 |

17 | 23818563824 | 3.7509 | 12.0112 | 91.7396 |

18 | 17638216924 | 2.7776 | 8.2604 | 94.5173 |

19 | 12512959412 | 1.9705 | 5.4827 | 96.4878 |

20 | 8546506844 | 1.3459 | 3.5122 | 97.8336 |

21 | 5599172640 | 0.8817 | 2.1664 | 98.7154 |

22 | 3509161412 | 0.5526 | 1.2846 | 99.2680 |

23 | 2112451680 | 0.3327 | 0.7320 | 99.6007 |

24 | 1215549644 | 0.1914 | 0.3993 | 99.7921 |

25 | 665258884 | 0.1048 | 0.2079 | 99.8968 |

26 | 346309380 | 0.0545 | 0.1032 | 99.9514 |

27 | 171594884 | 0.0270 | 0.0486 | 99.9784 |

28 | 79694360 | 0.0126 | 0.0216 | 99.9909 |

29 | 35006360 | 0.0055 | 0.0091 | 99.9965 |

30 | 14392632 | 0.0023 | 0.0035 | 99.9987 |

31 | 5429168 | 0.0009 | 0.0013 | 99.9996 |

32 | 1895604 | 0.0003 | 0.0004 | 99.9999 |

33 | 574548 | 0.0001 | 0.0001 | 100.0000 |

34 | 153448 | 0.0000 | 0.0000 | 100.0000 |

35 | 31924 | 0.0000 | 0.0000 | 100.0000 |

36 | 5172 | 0.0000 | 0.0000 | 100.0000 |

37 | 508 | 0.0000 | 0.0000 | 100.0000 |

38 | 16 | 0.0000 | 0.0000 | 100 |

U = 39 | 635013559600 | 100 | 4000 |

Playing tricks is a measurement of trick-taking potential with your longest suit trumps. This is typically used when you have a 6+ card suit, such as for a preemptive bid, but can be applied any time. The three highest 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 |

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 |

Quick tricks, or defensive tricks, is a measurement of defensive prospects with your shortest suit trumps. At most 2 tricks can be counted in each suit using the formula: A-K = 2, A-Q = 1.5, A or K-Q = 1, and K = 0.5. Note that a king does not have to be guarded, as a singleton king usually has the same half chance to win a trick.

Refinements: In a 7 or 8-card suit only *one* quick 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 quick 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 quick tricks, since a third trick is assured in whichever suit is trumps; similarly, at least A-K-J in every suit is arguably 8.5 quick tricks. These flukes are ignored; i.e., the maximum quick tricks per suit is always 2.

Quick 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 quick 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 quick 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 |

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 |

Guards is a measurement I devised to define stoppers for notrump play. A suit is considered 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. A suit is considered *partially* guarded — or in bridge parlance, half a stopper — if at least as good as K, Q-x, J-x-x, 10-x-x-x or any five cards. Each guarded suit = 2, and each partially guarded suit = 1. These values are exclusive (a suit cannot be both guarded and partially guarded) so the range for a hand is 0-8.

Guards Summary | |
---|---|

Hands evaluated | 635013559600 |

Unique values | 9 |

Minimum | 0 |

Maximum | 8 |

Mode | 6 |

Median | 5 |

Mean | 5.1750 |

Standard deviation | 1.4763 |

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., 5-6 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 |

1 | 1282854496 | 0.2020 | 99.9365 | 0.2655 |

2 | 27288313108 | 4.2973 | 99.7345 | 4.5628 |

3 | 34260614296 | 5.3953 | 95.4372 | 9.9581 |

4 | 172473892376 | 27.1607 | 90.0419 | 37.1187 |

5 | 98440762332 | 15.5022 | 62.8813 | 52.6209 |

6 | 208548676848 | 32.8416 | 47.3791 | 85.4625 |

7 | 44331564684 | 6.9812 | 14.5375 | 92.4437 |

8 | 47983568340 | 7.5563 | 7.5563 | 100 |

U = 9 | 635013559600 | 100 | 1000 |

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 |

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 |

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.

© 2012 Richard Pavlicek