Study 8J19 Main

Kaplan-Rubens Evaluation Stats

by Richard Pavlicek

In the early 1980s Edgar Kaplan devised a hand-evaluation method with various tweaks, attempting to translate expert judgment into a rigid formula. Jeff Rubens provided the mathematical expertise and oversight to fine-tune the project. While too complicated for practical use, it is often implemented by computer, and even to this day is highly regarded as the definitive gauge to evaluate a bridge hand. The algorithm is outlined below in 26 steps, applied to each suit.

1Ace4
2King3
3Queen2
4Jack1
5Ten0.5
62-6 cards with ten accompanied by jack or two (or more) higher honors0.5
72-6 cards with nine accompanied by eight, ten, or exactly two higher honors0.5
84-6 cards with nine (no eight or ten) and exactly three higher honors0.5
97+ cards missing queen or jack (or both)1
108+ cards missing queen1
119+ cards missing queen and jack1
12Multiply current total by suit length, divide by 10, then continue
13Ace3
14King guarded2
15King singleton0.5
16Queen (3+ cards) with ace or king (or both)1
17Queen (3+ cards) with no higher honor0.75
18Queen doubleton with ace or king0.5
19Queen doubleton with no higher honor0.25
20Jack with exactly two higher honors0.5
21Jack with exactly one higher honor0.25
22Ten with exactly two higher honors0.25
23Ten and nine with exactly one higher honor0.25
24Void3
25Singleton2
26Doubleton1

 Finally, add all four suit totals and subtract 1, then if 4-3-3-3 add 0.5

A few rules, particularly Step 8, are ambiguously explained by The Bridge World, so I have implemented what I think was Edgar Kaplan’s intent — to which I have first-hand experience from the old days. Playing on teams with Edgar, at dinner break he would sometimes pull out his trusty calculator and punch in a hand that I overbid and went set, carefully noting that my 13-point opener was really worth only 11.xx. When I asked if they opened at his table he would dryly admit, “Yes… That’s why we’re ahead; they can’t bid either.”

To evaluate a specific hand
Bridge Hand Evaluator

Curiously, Kaplan-Rubens points can be negative, e.g., a 4-3-3-3 Yarborough evaluates to minus 0.5. While I don’t particularly like this — zero-basing all methods seems right for uniformity — I won’t disturb tradition by trying to change it. Maybe this was Edgar’s way of taming his partners. Norman Kay, after bidding 3 NT on his own: “Got anything, pard?” Edgar: “No, it’s worse than that.”

Kaplan-Rubens Evaluation Summary
Hands evaluated635,013,559,600
Unique values710
Minimum-0.50
Maximum35.60
Mode11.20
Median10.65
Mean10.7725
Standard deviation4.7233

Every possible bridge hand was evaluated to compile the data. K-R points range from -0.5 to 35.6 in 0.05 increments, noncontiguous at the extremes. Impossible values are -0.45, -0.4, -0.25, 34.45, 34.75, 34.95, 35, 35.15, 35.25, 35.35, 35.45, 35.5, and 35.55. The maximum K-R value of 35.6 is amusing, as it represents the 7 NT claimer (pictured below) that everyone else counts as 37 (or 38 if four aces get a point). I wish I had noticed this years ago, as I’d love to hear Edgar’s explanation for the reduction.

 Study 8J19 Main Top Kaplan-Rubens Evaluation Stats

Specific Occurrences

The following table shows the number of hands and percent chance to the nearest 10,000th for each number of K-R 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, add the specific percents for each value in that range; or for exact precision, add the number of hands and divide by 6350135596.

PointsHandsSpecificAt LeastAt Most
-0.5027,500,0000.00431000.0043
-0.3544,550,0000.007099.99570.0113
-0.3025,000,0000.003999.98870.0153
-0.2080,157,0000.012699.98470.0279
-0.1540,500,0000.006499.97210.0343
-0.1035,500,0000.005699.96570.0399
-0.0566,568,2600.010599.96010.0504
0151,035,0000.023899.94960.0741
0.0557,510,0000.009199.92590.0832
0.1093,998,7600.014899.91680.0980
0.15118,472,1000.018799.90200.1167
0.20218,630,4000.034499.88330.1511
0.25102,381,0600.016199.84890.1672
0.30180,055,6560.028499.83280.1956
0.35166,566,0720.026299.80440.2218
0.40309,586,4440.048899.77820.2705
0.45102,222,2040.016199.72950.2866
0.50313,617,0240.049499.71340.3360
0.55205,923,4680.032499.66400.3685
0.60326,354,5400.051499.63150.4199
0.65209,697,8400.033099.58010.4529
0.70257,600,4000.040699.54710.4934
0.75206,117,8160.032599.50660.5259
0.80422,432,4440.066599.47410.5924
0.85168,005,5000.026599.40760.6189
0.90291,628,3800.045999.38110.6648
0.95212,836,8600.033599.33520.6983
1367,898,2760.057999.30170.7563
1.05238,465,3480.037699.24370.7938
1.10255,477,1960.040299.20620.8340
1.15271,785,2520.042899.16600.8768
1.20406,681,8880.064099.12320.9409
1.25247,237,6400.038999.05910.9798
1.30315,165,1000.049699.02021.0295
1.35383,013,1680.060398.97051.0898
1.40381,182,0040.060098.91021.1498
1.45314,501,7960.049598.85021.1993
1.50430,154,6960.067798.80071.2671
1.55475,385,9040.074998.73291.3419
1.60430,396,6480.067898.65811.4097
1.65450,626,5920.071098.59031.4807
1.70482,009,7120.075998.51931.5566
1.75608,776,9200.095998.44341.6524
1.80448,383,4560.070698.34761.7230
1.85544,818,5520.085898.27701.8088
1.90599,814,4400.094598.19121.9033
1.95644,919,3920.101698.09672.0049
2565,317,4440.089097.99512.0939
2.05514,179,9960.081097.90612.1749
2.10524,616,3160.082697.82512.2575
2.15638,769,0960.100697.74252.3581
2.20551,351,9200.086897.64192.4449
2.25497,117,8400.078397.55512.5232
2.30533,676,5720.084097.47682.6072
2.35544,642,9880.085897.39282.6930
2.40640,553,4120.100997.30702.7939
2.45356,760,1560.056297.20612.8500
2.50593,658,7720.093597.15002.9435
2.55643,866,0000.101497.05653.0449
2.60622,118,6440.098096.95513.1429
2.65418,062,1320.065896.85713.2087
2.70692,391,9360.109096.79133.3178
2.75665,340,7920.104896.68223.4225
2.80663,120,4600.104496.57753.5270
2.85553,622,9600.087296.47303.6141
2.90759,471,2960.119696.38593.7337
2.95699,715,2240.110296.26633.8439
3834,916,1400.131596.15613.9754
3.05692,111,6560.109096.02464.0844
3.10883,344,2280.139195.91564.2235
3.15792,512,2560.124895.77654.3483
3.20868,136,0760.136795.65174.4850
3.25823,320,7760.129795.51504.6147
3.30927,225,3000.146095.38534.7607
3.35786,533,6560.123995.23934.8846
3.401,077,032,6520.169695.11545.0542
3.45691,405,5520.108994.94585.1630
3.501,014,691,7680.159894.83705.3228
3.55878,102,7680.138394.67725.4611
3.60898,885,5320.141694.53895.6027
3.65773,365,1280.121894.39735.7245
3.701,027,612,5160.161894.27555.8863
3.75932,550,0760.146994.11376.0331
3.80958,212,0120.150993.96696.1840
3.85863,001,1560.135993.81606.3199
3.901,088,969,9640.171593.68016.4914
3.95971,477,3760.153093.50866.6444
4939,874,6480.148093.35566.7924
4.051,048,086,3760.165093.20766.9575
4.101,133,632,6720.178593.04257.1360
4.15985,014,2680.155192.86407.2911
4.201,147,101,6960.180692.70897.4717
4.251,223,907,6800.192792.52837.6645
4.301,063,474,3720.167592.33557.8320
4.351,100,257,3720.173392.16808.0052
4.401,325,101,8560.208791.99488.2139
4.451,166,489,3560.183791.78618.3976
4.501,244,043,8240.195991.60248.5935
4.551,297,660,7520.204491.40658.7979
4.601,404,246,7960.221191.20219.0190
4.651,208,647,9600.190390.98109.2093
4.701,391,967,4840.219290.79079.4285
4.751,466,668,5880.231090.57159.6595
4.801,480,934,7520.233290.34059.8927
4.851,349,115,2680.212590.107310.1052
4.901,439,233,1280.226689.894810.3318
4.951,450,625,4080.228489.668210.5602

PointsHandsSpecificAt LeastAt Most
51,686,992,7240.265789.439810.8259
5.051,315,984,7640.207289.174111.0331
5.101,567,177,9600.246888.966911.2799
5.151,419,748,8320.223688.720111.5035
5.201,606,184,4400.252988.496511.7565
5.251,568,001,3080.246988.243512.0034
5.301,612,077,2440.253987.996612.2572
5.351,344,021,9880.211787.742812.4689
5.401,878,305,0160.295887.531112.7647
5.451,467,982,6520.231287.235312.9959
5.501,565,876,0400.246687.004113.2424
5.551,564,242,7000.246386.757613.4888
5.601,821,869,3600.286986.511213.7757
5.651,569,437,6200.247286.224314.0228
5.701,652,586,9760.260285.977214.2831
5.751,797,969,9480.283185.716914.5662
5.801,748,012,8680.275385.433814.8415
5.851,647,811,4240.259585.158515.1010
5.901,729,577,2040.272484.899015.3733
5.951,818,477,4600.286484.626715.6597
61,858,411,5560.292784.340315.9524
6.051,876,422,2480.295584.047616.2479
6.101,787,806,6440.281583.752116.5294
6.151,946,535,0960.306583.470616.8359
6.201,865,889,1280.293883.164117.1298
6.251,840,187,6640.289882.870217.4196
6.301,925,996,9720.303382.580417.7229
6.351,957,217,7680.308282.277118.0311
6.402,069,736,7640.325981.968918.3570
6.451,739,071,8280.273981.643018.6309
6.502,013,536,1640.317181.369118.9480
6.551,820,883,1960.286781.052019.2347
6.602,103,003,3480.331280.765319.5659
6.651,783,196,1200.280880.434119.8467
6.702,047,398,1080.322480.153320.1691
6.751,951,452,7560.307379.830920.4764
6.802,288,682,4120.360479.523620.8368
6.851,807,145,3760.284679.163221.1214
6.902,016,872,5760.317678.878621.4390
6.952,046,642,6000.322378.561021.7613
72,329,960,1800.366978.238722.1282
7.051,882,830,9320.296577.871822.4248
7.102,071,380,0200.326277.575222.7509
7.152,218,514,3720.349477.249123.1003
7.202,049,820,5320.322876.899723.4231
7.252,030,932,7120.319876.576923.7429
7.302,187,928,7120.344576.257124.0875
7.352,167,673,2040.341475.912524.4288
7.402,155,953,2480.339575.571224.7684
7.452,165,195,8520.341075.231625.1093
7.502,196,777,9680.345974.890725.4553
7.552,278,106,4600.358774.544725.8140
7.602,343,902,6480.369174.186026.1831
7.652,035,758,7720.320673.816926.5037
7.702,379,814,6080.374873.496326.8785
7.752,302,810,3120.362673.121527.2411
7.802,290,664,4880.360772.758927.6018
7.852,166,624,5280.341272.398227.9430
7.902,383,072,6320.375372.057028.3183
7.952,124,974,3440.334671.681728.6529
82,682,577,4200.422471.347129.0754
8.052,042,136,7680.321670.924629.3970
8.102,382,568,8240.375270.603029.7722
8.152,411,064,7680.379770.227830.1519
8.202,451,496,9960.386169.848130.5379
8.252,216,633,4960.349169.462130.8870
8.302,619,438,0160.412569.113031.2995
8.352,300,090,0040.362268.700531.6617
8.402,425,422,1200.381968.338332.0437
8.452,464,654,8000.388167.956332.4318
8.502,527,111,9920.398067.568232.8297
8.552,295,431,6360.361567.170333.1912
8.602,525,031,5240.397666.808833.5889
8.652,584,002,3640.406966.411133.9958
8.702,282,781,8880.359566.004234.3553
8.752,547,730,3920.401265.644734.7565
8.802,553,221,3560.402165.243535.1585
8.852,344,043,7680.369164.841535.5277
8.902,436,898,5640.383864.472335.9114
8.952,554,011,2360.402264.088636.3136
92,506,481,6840.394763.686436.7083
9.052,437,810,5720.383963.291737.0922
9.102,574,684,5440.405562.907837.4977
9.152,362,297,4560.372062.502337.8697
9.202,701,100,3960.425462.130338.2951
9.252,416,818,7760.380661.704938.6757
9.302,583,979,6560.406961.324339.0826
9.352,414,594,3720.380260.917439.4628
9.402,753,816,3320.433760.537239.8965
9.452,426,701,6040.382160.103540.2786
9.502,726,477,3320.429459.721440.7080
9.552,372,588,5160.373659.292041.0816
9.602,786,591,4920.438858.918441.5204
9.652,363,554,7640.372258.479641.8926
9.702,671,708,2360.420758.107442.3134
9.752,540,702,2920.400157.686642.7135
9.802,723,284,5000.428957.286543.1423
9.852,703,933,9760.425856.857743.5681
9.902,606,786,1560.410556.431943.9786
9.952,574,991,3520.405556.021444.3841

PointsHandsSpecificAt LeastAt Most
102,739,332,2720.431455.615944.8155
10.052,655,382,0520.418255.184545.2337
10.102,574,527,3720.405454.766345.6391
10.152,660,718,2040.419054.360946.0581
10.202,557,078,9880.402753.941946.4608
10.252,634,119,1400.414853.539246.8756
10.302,502,626,1640.394153.124447.2697
10.352,497,402,5680.393352.730347.6630
10.402,440,511,3080.384352.337048.0473
10.452,564,844,6200.403951.952748.4512
10.502,669,569,2160.420451.548848.8716
10.552,411,248,7640.379751.128449.2513
10.602,815,611,7720.443450.748749.6947
10.652,323,263,4600.365950.305350.0606
10.702,641,362,1600.416049.939450.4766
10.752,511,085,3480.395449.523450.8720
10.802,707,583,1280.426449.128051.2984
10.852,316,428,8440.364848.701651.6632
10.902,791,447,7800.439648.336852.1027
10.952,253,079,1360.354847.897352.4576
112,806,684,1600.442047.542452.8995
11.052,404,495,3320.378747.100553.2782
11.102,506,036,5920.394646.721853.6728
11.152,402,737,6720.378446.327254.0512
11.202,817,112,9240.443645.948854.4948
11.252,532,254,8520.398845.505254.8936
11.302,446,405,5720.385345.106455.2789
11.352,716,100,9880.427744.721155.7066
11.402,558,501,0640.402944.293456.1095
11.452,453,749,7680.386443.890556.4959
11.502,653,624,6320.417943.504156.9138
11.552,548,427,5840.401343.086257.3151
11.602,398,493,7320.377742.684957.6928
11.652,594,252,3160.408542.307258.1014
11.702,423,502,2680.381641.898658.4830
11.752,413,875,8560.380141.517058.8631
11.802,499,958,8000.393741.136959.2568
11.852,369,000,9240.373140.743259.6299
11.902,448,854,7000.385640.370160.0155
11.952,317,505,6120.365039.984560.3805
122,589,655,3880.407839.619560.7883
12.052,287,816,2200.360339.211761.1486
12.102,557,864,0680.402838.851461.5514
12.152,197,616,5160.346138.448661.8974
12.202,521,288,9840.397038.102662.2945
12.252,226,369,1040.350637.705562.6451
12.302,479,222,6400.390437.354963.0355
12.352,260,753,7400.356036.964563.3915
12.402,506,551,6200.394736.608563.7862
12.452,141,708,7280.337336.213864.1235
12.502,476,664,7040.390035.876564.5135
12.552,270,739,0440.357635.486564.8711
12.602,493,161,8760.392635.128965.2637
12.652,251,130,9240.354534.736365.6182
12.702,404,354,4880.378634.381865.9969
12.752,345,708,0680.369434.003166.3663
12.802,350,902,6360.370233.633766.7365
12.852,364,630,7360.372433.263567.1089
12.902,265,056,3760.356732.891167.4655
12.952,182,202,4680.343632.534567.8092
132,411,457,3680.379732.190868.1889
13.052,203,233,4920.347031.811168.5359
13.102,188,709,5200.344731.464168.8806
13.152,152,523,2040.339031.119469.2195
13.202,192,382,2760.345230.780569.5648
13.252,187,162,8600.344430.435269.9092
13.302,221,897,1280.349930.090870.2591
13.352,073,079,9400.326529.740970.5856
13.402,221,708,7640.349929.414470.9354
13.451,984,786,8520.312629.064671.2480
13.502,231,559,8280.351428.752071.5994
13.551,962,110,6720.309028.400671.9084
13.602,181,195,5000.343528.091672.2519
13.652,022,232,7400.318527.748172.5704
13.702,076,140,1760.326927.429672.8973
13.751,997,659,9160.314627.102773.2119
13.802,157,356,3920.339726.788173.5516
13.851,833,783,2200.288826.448473.8404
13.902,146,992,4360.338126.159674.1785
13.952,003,897,8160.315625.821574.4941
142,069,788,8120.325925.505974.8200
14.051,930,222,5520.304025.180075.1240
14.102,118,682,5280.333624.876075.4576
14.151,857,456,7800.292524.542475.7501
14.202,019,882,0840.318124.249976.0682
14.251,972,481,6160.310623.931876.3788
14.301,953,448,6640.307623.621276.6865
14.351,832,249,6200.288523.313576.9750
14.402,032,300,7720.320023.025077.2950
14.451,825,964,6440.287522.705077.5826
14.501,884,580,7560.296822.417477.8794
14.551,782,774,5320.280722.120678.1601
14.601,841,187,8320.289921.839978.4500
14.651,767,828,7440.278421.550078.7284
14.701,867,547,0480.294121.271679.0225
14.751,665,704,0400.262320.977579.2848
14.801,775,069,4000.279520.715279.5644
14.851,699,318,3400.267620.435679.8320
14.901,730,005,2400.272420.168080.1044
14.951,591,301,7760.250619.895680.3550

PointsHandsSpecificAt LeastAt Most
151,814,224,9440.285719.645080.6407
15.051,593,324,8240.250919.359380.8916
15.101,739,708,6440.274019.108481.1656
15.151,555,245,6400.244918.834481.4105
15.201,714,106,7280.269918.589581.6804
15.251,595,358,0200.251218.319681.9317
15.301,720,194,0520.270918.068382.2026
15.351,581,340,0200.249017.797482.4516
15.401,654,765,4920.260617.548482.7122
15.451,506,769,2320.237317.287882.9495
15.501,678,710,6720.264417.050583.2138
15.551,524,055,4720.240016.786283.4538
15.601,643,567,0240.258816.546283.7126
15.651,428,043,0720.224916.287483.9375
15.701,547,280,0720.243716.062584.1812
15.751,524,945,4960.240115.818884.4213
15.801,513,151,0240.238315.578784.6596
15.851,459,214,0960.229815.340484.8894
15.901,473,311,9560.232015.110685.1214
15.951,362,681,6560.214614.878685.3360
161,469,608,0960.231414.664085.5674
16.051,360,018,6320.214214.432685.7816
16.101,414,776,4920.222814.218486.0044
16.151,324,492,0280.208613.995686.2130
16.201,382,160,6080.217713.787086.4306
16.251,308,314,8400.206013.569486.6367
16.301,359,156,4800.214013.363386.8507
16.351,246,390,7280.196313.149387.0470
16.401,339,061,6440.210912.953087.2579
16.451,175,265,8680.185112.742187.4429
16.501,383,313,0480.217812.557187.6608
16.551,174,061,9080.184912.339287.8457
16.601,302,289,2000.205112.154388.0507
16.651,152,464,6320.181511.949388.2322
16.701,262,198,0600.198811.767888.4310
16.751,137,870,3120.179211.569088.6102
16.801,277,218,8400.201111.389888.8113
16.851,120,273,7160.176411.188788.9877
16.901,231,572,2000.193911.012389.1817
16.951,083,928,8520.170710.818389.3524
171,221,136,5480.192310.647689.5447
17.051,068,160,3600.168210.455389.7129
17.101,148,548,1000.180910.287189.8938
17.151,063,154,6200.167410.106290.0612
17.201,131,178,1960.17819.938890.2393
17.251,034,148,8200.16299.760790.4022
17.301,088,012,4360.17139.597890.5735
17.35991,393,4600.15619.426590.7296
17.401,072,218,3200.16889.270490.8985
17.45986,516,6160.15549.101591.0538
17.501,021,442,5480.16098.946291.2147
17.55936,931,1960.14758.785391.3622
17.601,063,506,1360.16758.637891.5297
17.65901,242,8240.14198.470391.6716
17.70957,009,2360.15078.328491.8223
17.75920,533,9440.14508.177791.9673
17.80946,088,5800.14908.032792.1163
17.85864,720,7800.13627.883792.2525
17.90975,912,4720.15377.747592.4061
17.95810,282,3960.12767.593992.5337
18927,769,0360.14617.466392.6798
18.05799,792,6480.12597.320292.8058
18.10912,164,5200.14367.194292.9494
18.15793,569,5840.12507.050693.0744
18.20865,332,2040.13636.925693.2107
18.25789,779,6400.12446.789393.3350
18.30826,390,6440.13016.665093.4652
18.35757,788,2760.11936.534893.5845
18.40804,567,4280.12676.415593.7112
18.45714,221,3720.11256.288893.8237
18.50804,705,5600.12676.176393.9504
18.55717,462,4080.11306.049694.0634
18.60760,722,6680.11985.936694.1832
18.65690,050,0600.10875.816894.2919
18.70706,304,3240.11125.708194.4031
18.75682,645,5280.10755.596994.5106
18.80745,078,9440.11735.489494.6279
18.85617,550,3560.09725.372194.7252
18.90692,538,8680.10915.274894.8342
18.95620,896,0840.09785.165894.9320
19684,502,2400.10785.068095.0398
19.05611,759,7240.09634.960295.1361
19.10643,944,5160.10144.863995.2375
19.15580,434,2400.09144.762595.3290
19.20642,247,3080.10114.671095.4301
19.25575,866,2280.09074.569995.5208
19.30624,123,8800.09834.479295.6191
19.35531,126,3960.08364.380995.7027
19.40614,316,3520.09674.297395.7994
19.45525,218,2000.08274.200695.8822
19.50605,372,7200.09534.117895.9775
19.55520,194,6040.08194.022596.0594
19.60556,740,4520.08773.940696.1471
19.65502,107,9640.07913.852996.2261
19.70552,542,0600.08703.773996.3132
19.75470,776,0520.07413.686896.3873
19.80531,970,0000.08383.612796.4711
19.85458,214,4280.07223.528996.5432
19.90496,023,9560.07813.456896.6213
19.95444,272,3920.07003.378796.6913

PointsHandsSpecificAt LeastAt Most
20488,506,0280.07693.308796.7682
20.05426,333,0880.06713.231896.8354
20.10474,974,6920.07483.164696.9102
20.15419,486,5880.06613.089896.9762
20.20455,794,5280.07183.023897.0480
20.25388,332,2160.06122.952097.1092
20.30446,237,4320.07032.890897.1794
20.35394,165,3400.06212.820697.2415
20.40414,112,6920.06522.758597.3067
20.45362,960,6280.05722.693397.3639
20.50403,234,0760.06352.636197.4274
20.55350,654,6240.05522.572697.4826
20.60397,575,1760.06262.517497.5452
20.65341,377,4840.05382.454897.5990
20.70367,480,5560.05792.401097.6568
20.75327,784,1920.05162.343297.7084
20.80377,942,6080.05952.291697.7680
20.85316,514,7840.04982.232097.8178
20.90342,357,0800.05392.182297.8717
20.95299,432,6200.04722.128397.9189
21348,052,3800.05482.081197.9737
21.05283,806,8160.04472.026398.0184
21.10338,242,6800.05331.981698.0716
21.15266,519,5360.04201.928498.1136
21.20321,234,9520.05061.886498.1642
21.25265,842,2560.04191.835898.2061
21.30297,160,5040.04681.793998.2529
21.35254,382,2520.04011.747198.2929
21.40303,215,7600.04771.707198.3407
21.45232,017,3360.03651.659398.3772
21.50273,802,7200.04311.622898.4203
21.55233,883,5040.03681.579798.4572
21.60265,521,8160.04181.542898.4990
21.65231,071,6480.03641.501098.5354
21.70249,310,9920.03931.464698.5746
21.75213,197,8440.03361.425498.6082
21.80255,197,7120.04021.391898.6484
21.85206,124,3200.03251.351698.6808
21.90226,879,0280.03571.319298.7166
21.95202,827,5200.03191.283498.7485
22224,370,8160.03531.251598.7838
22.05190,848,4400.03011.216298.8139
22.10215,306,9360.03391.186198.8478
22.15181,037,1200.02851.152298.8763
22.20199,057,9560.03131.123798.9077
22.25179,244,9480.02821.092398.9359
22.30191,144,7040.03011.064198.9660
22.35160,099,9240.02521.034098.9912
22.40190,549,7800.03001.008899.0212
22.45160,849,7200.02530.978899.0465
22.50171,944,4120.02710.953599.0736
22.55146,371,7000.02310.926499.0967
22.60175,518,1000.02760.903399.1243
22.65137,548,4840.02170.875799.1460
22.70164,890,3000.02600.854099.1719
22.75132,869,2560.02090.828199.1929
22.80154,515,1680.02430.807199.2172
22.85127,254,3360.02000.782899.2372
22.90143,767,7640.02260.762899.2599
22.95118,942,7800.01870.740199.2786
23151,260,7360.02380.721499.3024
23.05112,706,4080.01770.697699.3202
23.10134,904,4520.02120.679899.3414
23.15111,675,7800.01760.658699.3590
23.20129,522,8880.02040.641099.3794
23.25104,966,1160.01650.620699.3959
23.30120,934,9600.01900.604199.4150
23.35101,568,5080.01600.585099.4310
23.40117,170,2040.01850.569099.4494
23.4588,748,1200.01400.550699.4634
23.50110,770,1160.01740.536699.4808
23.5589,708,9200.01410.519299.4950
23.60106,116,8880.01670.505099.5117
23.6586,438,9960.01360.488399.5253
23.7099,129,2800.01560.474799.5409
23.7579,511,5200.01250.459199.5534
23.8097,853,7040.01540.446699.5688
23.8581,282,0840.01280.431299.5816
23.9090,019,9320.01420.418499.5958
23.9569,061,2600.01090.404299.6067
2489,930,2280.01420.393399.6208
24.0567,220,9400.01060.379299.6314
24.1078,122,2880.01230.368699.6437
24.1565,541,8520.01030.356399.6540
24.2074,442,9960.01170.346099.6658
24.2560,475,7360.00950.334299.6753
24.3073,401,6160.01160.324799.6869
24.3555,797,3120.00880.313199.6956
24.4070,318,4520.01110.304499.7067
24.4553,768,0400.00850.293399.7152
24.5064,298,9360.01010.284899.7253
24.5552,121,1040.00820.274799.7335
24.6065,011,6120.01020.266599.7438
24.6547,514,0680.00750.256299.7512
24.7058,759,1720.00930.248899.7605
24.7543,274,1720.00680.239599.7673
24.8057,189,0840.00900.232799.7763
24.8541,217,1360.00650.223799.7828
24.9052,279,3600.00820.217299.7910
24.9538,316,5520.00600.209099.7971

PointsHandsSpecificAt LeastAt Most
2549,002,2640.00770.202999.8048
25.0539,923,0600.00630.195299.8111
25.1043,238,4200.00680.188999.8179
25.1535,790,4400.00560.182199.8235
25.2043,285,5440.00680.176599.8303
25.2530,424,0920.00480.169799.8351
25.3039,945,0360.00630.164999.8414
25.3531,322,5120.00490.158699.8463
25.4037,606,2120.00590.153799.8523
25.4529,942,4280.00470.147799.8570
25.5035,485,5280.00560.143099.8626
25.5527,069,0720.00430.137499.8668
25.6035,967,9480.00570.133299.8725
25.6525,309,8200.00400.127599.8765
25.7032,393,2560.00510.123599.8816
25.7522,854,6640.00360.118499.8852
25.8030,100,5000.00470.114899.8899
25.8523,010,1600.00360.110199.8935
25.9029,816,9320.00470.106599.8982
25.9520,620,1400.00320.101899.9015
2626,380,4400.00420.098599.9056
26.0519,204,9960.00300.094499.9087
26.1024,902,4480.00390.091399.9126
26.1517,346,0160.00270.087499.9153
26.2024,397,3800.00380.084799.9192
26.2517,311,0600.00270.080899.9219
26.3021,286,0520.00340.078199.9252
26.3515,933,4920.00250.074899.9278
26.4021,991,2240.00350.072299.9312
26.4514,804,4520.00230.068899.9335
26.5018,774,1080.00300.066599.9365
26.5513,837,4320.00220.063599.9387
26.6017,655,9840.00280.061399.9415
26.6513,056,4600.00210.058599.9435
26.7015,700,7720.00250.056599.9460
26.7511,992,5240.00190.054099.9479
26.8014,603,1160.00230.052199.9502
26.8511,336,5720.00180.049899.9520
26.9013,798,4000.00220.048099.9541
26.959,969,8960.00160.045999.9557
2711,962,9320.00190.044399.9576
27.059,764,9560.00150.042499.9591
27.1012,514,2400.00200.040999.9611
27.158,192,1960.00130.038999.9624
27.2012,434,3640.00200.037699.9643
27.257,070,1560.00110.035799.9655
27.3010,863,5640.00170.034599.9672
27.356,631,8520.00100.032899.9682
27.4010,788,4120.00170.031899.9699
27.456,480,0880.00100.030199.9709
27.509,704,8840.00150.029199.9725
27.556,029,9960.00090.027599.9734
27.608,575,2600.00140.026699.9748
27.655,723,3040.00090.025299.9757
27.707,962,8600.00130.024399.9769
27.755,350,3920.00080.023199.9778
27.807,450,6480.00120.022299.9789
27.855,159,2880.00080.021199.9797
27.906,254,9720.00100.020399.9807
27.954,898,4040.00080.019399.9815
287,098,6560.00110.018599.9826
28.054,282,4560.00070.017499.9833
28.105,638,6520.00090.016799.9842
28.153,708,5320.00060.015899.9848
28.205,769,7600.00090.015299.9857
28.253,569,4560.00060.014399.9862
28.305,018,1400.00080.013899.9870
28.353,407,1760.00050.013099.9876
28.404,257,9240.00070.012499.9882
28.453,007,1120.00050.011899.9887
28.504,071,8200.00060.011399.9894
28.552,524,6880.00040.010699.9897
28.603,610,1040.00060.010399.9903
28.652,225,3080.00040.009799.9907
28.703,590,1000.00060.009399.9912
28.751,854,4640.00030.008899.9915
28.802,922,0120.00050.008599.9920
28.852,151,2800.00030.008099.9923
28.902,984,2760.00050.007799.9928
28.951,732,5640.00030.007299.9931
292,873,1160.00050.006999.9935
29.051,497,8000.00020.006599.9938
29.102,400,5680.00040.006299.9941
29.151,573,8840.00020.005999.9944
29.202,213,5920.00030.005699.9947
29.251,275,3080.00020.005399.9949
29.302,118,8440.00030.005199.9953
29.351,254,1320.00020.004799.9955
29.402,065,2120.00030.004599.9958
29.451,192,3040.00020.004299.9960
29.501,494,9480.00020.004099.9962
29.551,017,4560.00020.003899.9964
29.601,501,3680.00020.003699.9966
29.651,050,3640.00020.003499.9968
29.701,336,7520.00020.003299.9970
29.75920,5520.00010.003099.9971
29.801,303,1040.00020.002999.9973
29.85622,0360.00010.002799.9974
29.901,231,7040.00020.002699.9976
29.95590,6720.00010.002499.9977

PointsHandsSpecificAt LeastAt Most
301,059,7840.00020.002399.9979
30.05582,4080.00010.002199.9980
30.10888,9240.00010.002099.9981
30.15526,6880.00010.001999.9982
30.20900,5880.00010.001899.9983
30.25423,3040.00010.001799.9984
30.30764,2360.00010.001699.9985
30.35416,5880.00010.001599.9986
30.40631,4520.00010.001499.9987
30.45372,3600.00010.001399.9988
30.50532,5720.00010.001299.9988
30.55382,8360.00010.001299.9989
30.60609,1480.00010.001199.9990
30.65273,0000.00000.001099.9990
30.70477,0520.00010.001099.9991
30.75201,1320.00000.000999.9991
30.80503,6400.00010.000999.9992
30.85232,7120.00000.000899.9993
30.90466,8080.00010.000799.9993
30.95220,2760.00000.000799.9994
31243,7000.00000.000699.9994
31.05181,8880.00000.000699.9994
31.10327,6760.00010.000699.9995
31.15178,7440.00000.000599.9995
31.20246,9080.00000.000599.9996
31.25158,4000.00000.000499.9996
31.30250,0200.00000.000499.9996
31.3578,5440.00000.000499.9996
31.40273,1320.00000.000499.9997
31.4594,2880.00000.000399.9997
31.50122,7200.00000.000399.9997
31.5593,4600.00000.000399.9997
31.60183,1800.00000.000399.9997
31.6560,8560.00000.000399.9998
31.70145,8360.00000.000299.9998
31.7581,1680.00000.000299.9998
31.80134,8960.00000.000299.9998
31.8545,2920.00000.000299.9998
31.90127,9160.00000.000299.9998
31.9549,7640.00000.000299.9999
3294,8240.00000.000199.9999
32.0543,4080.00000.000199.9999
32.1072,0160.00000.000199.9999
32.1535,3280.00000.000199.9999
32.2082,0000.00000.000199.9999
32.2544,2360.00000.000199.9999
32.3072,4840.00000.000199.9999
32.3529,9400.00000.000199.9999
32.4063,3520.00000.000199.9999
32.4519,8560.00000.000199.9999
32.5043,5960.00000.000199.9999
32.5523,4960.00000.000199.9999
32.6027,5880.00000.0001100.0000
32.6513,7320.00000.0000100.0000
32.7036,5760.00000.0000100.0000
32.7520,1480.00000.0000100.0000
32.8041,9080.00000.0000100.0000
32.8574320.00000.0000100.0000
32.9020,7520.00000.0000100.0000
32.9552680.00000.0000100.0000
3311,3480.00000.0000100.0000
33.0510,4760.00000.0000100.0000
33.1014,7760.00000.0000100.0000
33.1553640.00000.0000100.0000
33.2014,4720.00000.0000100.0000
33.2538280.00000.0000100.0000
33.3013,0360.00000.0000100.0000
33.3517600.00000.0000100.0000
33.4013,6680.00000.0000100.0000
33.4525920.00000.0000100.0000
33.5084000.00000.0000100.0000
33.5538040.00000.0000100.0000
33.6064960.00000.0000100.0000
33.659360.00000.0000100.0000
33.7053160.00000.0000100.0000
33.759720.00000.0000100.0000
33.8076480.00000.0000100.0000
33.8524360.00000.0000100.0000
33.9033840.00000.0000100.0000
33.952280.00000.0000100.0000
3423640.00000.0000100.0000
34.057240.00000.0000100.0000
34.1015000.00000.0000100.0000
34.157320.00000.0000100.0000
34.2012120.00000.0000100.0000
34.25520.00000.0000100.0000
34.3020440.00000.0000100.0000
34.351560.00000.0000100.0000
34.4018320.00000.0000100.0000
34.507960.00000.0000100.0000
34.556360.00000.0000100.0000
34.602520.00000.0000100.0000
34.65840.00000.0000100.0000
34.704120.00000.0000100.0000
34.806720.00000.0000100.0000
34.85120.00000.0000100.0000
34.901320.00000.0000100.0000
35.05360.00000.0000100.0000
35.101440.00000.0000100.0000
35.20280.00000.0000100.0000
35.30360.00000.0000100.0000
35.40160.00000.0000100.0000
35.6080.00000.0000100
U = 710635,013,559,60010071,100

Percents shown as 0.0000 are not zero but round to zero to four decimal places. Likewise, 100.0000 is not the same as 100 but falls short by a negligible amount. Check total of At Least + At Most should be 100(U+1).

 Study 8J19 Main Top Kaplan-Rubens Evaluation Stats