Main Study 8J17 by Richard Pavlicek

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 HCP Points Playing Tricks Losing Tricks Quick Tricks Controls Guards Freakness Longest Shortest *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

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 |

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 |

K, Q-x, K-x, J-10-x, Q-x-x | 0.5 | A-K, K-Q-J, A-Q-10, A-K-x | 2 |

A, K-J, K-Q, A-x, Q-J-x, K-x-x, A-x-x | 1 | A-Q-J, A-K-J | 2.5 |

A-J, A-Q, K-J-10, K-Q-x, A-J-x, A-Q-x | 1.5 | 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 |

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 |

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 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 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 |

4-3-3-3 = 0 | 5-4-3-1 = 4 | 6-4-2-1 = 6 | 6-5-1-1 = 8 | 8-2-2-1 = 9 | 8-4-1-0 = 11 | 8-5-0-0 = 13 | 11-1-1-0 = 16 |

4-4-3-2 = 1 | 6-3-2-2 = 4 | 7-2-2-2 = 6 | 7-3-3-0 = 8 | 8-3-1-1 = 9 | 9-2-1-1 = 11 | 9-4-0-0 = 13 | 11-2-0-0 = 16 |

5-3-3-2 = 2 | 6-3-3-1 = 5 | 5-5-3-0 = 7 | 7-4-1-1 = 8 | 8-3-2-0 = 10 | 9-2-2-0 = 12 | 10-1-1-1 = 13 | 12-1-0-0 = 18 |

4-4-4-1 = 3 | 5-4-4-0 = 6 | 6-4-3-0 = 7 | 6-5-2-0 = 9 | 6-6-1-0 = 11 | 9-3-1-0 = 12 | 10-2-1-0 = 14 | 13-0-0-0 = 20 |

5-4-2-2 = 3 | 5-5-2-1 = 6 | 7-3-2-1 = 7 | 7-4-2-0 = 9 | 7-5-1-0 = 11 | 7-6-0-0 = 13 | 10-3-0-0 = 14 |

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 |

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 |

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.

© 2015 Richard Pavlicek