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.

Step	Calculate Separately for Each Suit	Add
1	Ace	4
2	King	3
3	Queen	2
4	Jack	1
5	Ten	0.5
6	2-6 cards with ten accompanied by jack or two (or more) higher honors	0.5
7	2-6 cards with nine accompanied by eight, ten, or exactly two higher honors	0.5
8	4-6 cards with nine (no eight or ten) and exactly three higher honors	0.5
9	7+ cards missing queen or jack (or both)	1
10	8+ cards missing queen	1
11	9+ cards missing queen and jack	1
12	— Multiply current total by suit length, divide by 10, then continue —
13	Ace	3
14	King guarded	2
15	King singleton	0.5
16	Queen (3+ cards) with ace or king (or both)	1
17	Queen (3+ cards) with no higher honor	0.75
18	Queen doubleton with ace or king	0.5
19	Queen doubleton with no higher honor	0.25
20	Jack with exactly two higher honors	0.5
21	Jack with exactly one higher honor	0.25
22	Ten with exactly two higher honors	0.25
23	Ten and nine with exactly one higher honor	0.25
24	Void	3
25	Singleton	2
26	Doubleton	1

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 evaluated	635,013,559,600
Unique values	710
Minimum	-0.50
Maximum	35.60
Mode	11.20
Median	10.65
Mean	10.7725
Standard deviation	4.7233

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 tables show 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.

Percents of 0.0000 are not zero but round to zero to four places. Likewise, 100.0000 is not 100 but falls short by a small amount.

Points	Hands	Specific	At Least	At Most
-0.50	27,500,000	0.0043	100	0.0043
-0.35	44,550,000	0.0070	99.9957	0.0113
-0.30	25,000,000	0.0039	99.9887	0.0153
-0.20	80,157,000	0.0126	99.9847	0.0279
-0.15	40,500,000	0.0064	99.9721	0.0343
-0.10	35,500,000	0.0056	99.9657	0.0399
-0.05	66,568,260	0.0105	99.9601	0.0504
0	151,035,000	0.0238	99.9496	0.0741
0.05	57,510,000	0.0091	99.9259	0.0832
0.10	93,998,760	0.0148	99.9168	0.0980
0.15	118,472,100	0.0187	99.9020	0.1167
0.20	218,630,400	0.0344	99.8833	0.1511
0.25	102,381,060	0.0161	99.8489	0.1672
0.30	180,055,656	0.0284	99.8328	0.1956
0.35	166,566,072	0.0262	99.8044	0.2218
0.40	309,586,444	0.0488	99.7782	0.2705
0.45	102,222,204	0.0161	99.7295	0.2866
0.50	313,617,024	0.0494	99.7134	0.3360
0.55	205,923,468	0.0324	99.6640	0.3685
0.60	326,354,540	0.0514	99.6315	0.4199
0.65	209,697,840	0.0330	99.5801	0.4529
0.70	257,600,400	0.0406	99.5471	0.4934
0.75	206,117,816	0.0325	99.5066	0.5259
0.80	422,432,444	0.0665	99.4741	0.5924
0.85	168,005,500	0.0265	99.4076	0.6189
0.90	291,628,380	0.0459	99.3811	0.6648
0.95	212,836,860	0.0335	99.3352	0.6983

Points	Hands	Specific	At Least	At Most
1	367,898,276	0.0579	99.3017	0.7563
1.05	238,465,348	0.0376	99.2437	0.7938
1.10	255,477,196	0.0402	99.2062	0.8340
1.15	271,785,252	0.0428	99.1660	0.8768
1.20	406,681,888	0.0640	99.1232	0.9409
1.25	247,237,640	0.0389	99.0591	0.9798
1.30	315,165,100	0.0496	99.0202	1.0295
1.35	383,013,168	0.0603	98.9705	1.0898
1.40	381,182,004	0.0600	98.9102	1.1498
1.45	314,501,796	0.0495	98.8502	1.1993
1.50	430,154,696	0.0677	98.8007	1.2671
1.55	475,385,904	0.0749	98.7329	1.3419
1.60	430,396,648	0.0678	98.6581	1.4097
1.65	450,626,592	0.0710	98.5903	1.4807
1.70	482,009,712	0.0759	98.5193	1.5566
1.75	608,776,920	0.0959	98.4434	1.6524
1.80	448,383,456	0.0706	98.3476	1.7230
1.85	544,818,552	0.0858	98.2770	1.8088
1.90	599,814,440	0.0945	98.1912	1.9033
1.95	644,919,392	0.1016	98.0967	2.0049

Points	Hands	Specific	At Least	At Most
2	565,317,444	0.0890	97.9951	2.0939
2.05	514,179,996	0.0810	97.9061	2.1749
2.10	524,616,316	0.0826	97.8251	2.2575
2.15	638,769,096	0.1006	97.7425	2.3581
2.20	551,351,920	0.0868	97.6419	2.4449
2.25	497,117,840	0.0783	97.5551	2.5232
2.30	533,676,572	0.0840	97.4768	2.6072
2.35	544,642,988	0.0858	97.3928	2.6930
2.40	640,553,412	0.1009	97.3070	2.7939
2.45	356,760,156	0.0562	97.2061	2.8500
2.50	593,658,772	0.0935	97.1500	2.9435
2.55	643,866,000	0.1014	97.0565	3.0449
2.60	622,118,644	0.0980	96.9551	3.1429
2.65	418,062,132	0.0658	96.8571	3.2087
2.70	692,391,936	0.1090	96.7913	3.3178
2.75	665,340,792	0.1048	96.6822	3.4225
2.80	663,120,460	0.1044	96.5775	3.5270
2.85	553,622,960	0.0872	96.4730	3.6141
2.90	759,471,296	0.1196	96.3859	3.7337
2.95	699,715,224	0.1102	96.2663	3.8439

Points	Hands	Specific	At Least	At Most
3	834,916,140	0.1315	96.1561	3.9754
3.05	692,111,656	0.1090	96.0246	4.0844
3.10	883,344,228	0.1391	95.9156	4.2235
3.15	792,512,256	0.1248	95.7765	4.3483
3.20	868,136,076	0.1367	95.6517	4.4850
3.25	823,320,776	0.1297	95.5150	4.6147
3.30	927,225,300	0.1460	95.3853	4.7607
3.35	786,533,656	0.1239	95.2393	4.8846
3.40	1,077,032,652	0.1696	95.1154	5.0542
3.45	691,405,552	0.1089	94.9458	5.1630
3.50	1,014,691,768	0.1598	94.8370	5.3228
3.55	878,102,768	0.1383	94.6772	5.4611
3.60	898,885,532	0.1416	94.5389	5.6027
3.65	773,365,128	0.1218	94.3973	5.7245
3.70	1,027,612,516	0.1618	94.2755	5.8863
3.75	932,550,076	0.1469	94.1137	6.0331
3.80	958,212,012	0.1509	93.9669	6.1840
3.85	863,001,156	0.1359	93.8160	6.3199
3.90	1,088,969,964	0.1715	93.6801	6.4914
3.95	971,477,376	0.1530	93.5086	6.6444

Points	Hands	Specific	At Least	At Most
4	939,874,648	0.1480	93.3556	6.7924
4.05	1,048,086,376	0.1650	93.2076	6.9575
4.10	1,133,632,672	0.1785	93.0425	7.1360
4.15	985,014,268	0.1551	92.8640	7.2911
4.20	1,147,101,696	0.1806	92.7089	7.4717
4.25	1,223,907,680	0.1927	92.5283	7.6645
4.30	1,063,474,372	0.1675	92.3355	7.8320
4.35	1,100,257,372	0.1733	92.1680	8.0052
4.40	1,325,101,856	0.2087	91.9948	8.2139
4.45	1,166,489,356	0.1837	91.7861	8.3976
4.50	1,244,043,824	0.1959	91.6024	8.5935
4.55	1,297,660,752	0.2044	91.4065	8.7979
4.60	1,404,246,796	0.2211	91.2021	9.0190
4.65	1,208,647,960	0.1903	90.9810	9.2093
4.70	1,391,967,484	0.2192	90.7907	9.4285
4.75	1,466,668,588	0.2310	90.5715	9.6595
4.80	1,480,934,752	0.2332	90.3405	9.8927
4.85	1,349,115,268	0.2125	90.1073	10.1052
4.90	1,439,233,128	0.2266	89.8948	10.3318
4.95	1,450,625,408	0.2284	89.6682	10.5602

Points	Hands	Specific	At Least	At Most
5	1,686,992,724	0.2657	89.4398	10.8259
5.05	1,315,984,764	0.2072	89.1741	11.0331
5.10	1,567,177,960	0.2468	88.9669	11.2799
5.15	1,419,748,832	0.2236	88.7201	11.5035
5.20	1,606,184,440	0.2529	88.4965	11.7565
5.25	1,568,001,308	0.2469	88.2435	12.0034
5.30	1,612,077,244	0.2539	87.9966	12.2572
5.35	1,344,021,988	0.2117	87.7428	12.4689
5.40	1,878,305,016	0.2958	87.5311	12.7647
5.45	1,467,982,652	0.2312	87.2353	12.9959
5.50	1,565,876,040	0.2466	87.0041	13.2424
5.55	1,564,242,700	0.2463	86.7576	13.4888
5.60	1,821,869,360	0.2869	86.5112	13.7757
5.65	1,569,437,620	0.2472	86.2243	14.0228
5.70	1,652,586,976	0.2602	85.9772	14.2831
5.75	1,797,969,948	0.2831	85.7169	14.5662
5.80	1,748,012,868	0.2753	85.4338	14.8415
5.85	1,647,811,424	0.2595	85.1585	15.1010
5.90	1,729,577,204	0.2724	84.8990	15.3733
5.95	1,818,477,460	0.2864	84.6267	15.6597

Points	Hands	Specific	At Least	At Most
6	1,858,411,556	0.2927	84.3403	15.9524
6.05	1,876,422,248	0.2955	84.0476	16.2479
6.10	1,787,806,644	0.2815	83.7521	16.5294
6.15	1,946,535,096	0.3065	83.4706	16.8359
6.20	1,865,889,128	0.2938	83.1641	17.1298
6.25	1,840,187,664	0.2898	82.8702	17.4196
6.30	1,925,996,972	0.3033	82.5804	17.7229
6.35	1,957,217,768	0.3082	82.2771	18.0311
6.40	2,069,736,764	0.3259	81.9689	18.3570
6.45	1,739,071,828	0.2739	81.6430	18.6309
6.50	2,013,536,164	0.3171	81.3691	18.9480
6.55	1,820,883,196	0.2867	81.0520	19.2347
6.60	2,103,003,348	0.3312	80.7653	19.5659
6.65	1,783,196,120	0.2808	80.4341	19.8467
6.70	2,047,398,108	0.3224	80.1533	20.1691
6.75	1,951,452,756	0.3073	79.8309	20.4764
6.80	2,288,682,412	0.3604	79.5236	20.8368
6.85	1,807,145,376	0.2846	79.1632	21.1214
6.90	2,016,872,576	0.3176	78.8786	21.4390
6.95	2,046,642,600	0.3223	78.5610	21.7613

Points	Hands	Specific	At Least	At Most
7	2,329,960,180	0.3669	78.2387	22.1282
7.05	1,882,830,932	0.2965	77.8718	22.4248
7.10	2,071,380,020	0.3262	77.5752	22.7509
7.15	2,218,514,372	0.3494	77.2491	23.1003
7.20	2,049,820,532	0.3228	76.8997	23.4231
7.25	2,030,932,712	0.3198	76.5769	23.7429
7.30	2,187,928,712	0.3445	76.2571	24.0875
7.35	2,167,673,204	0.3414	75.9125	24.4288
7.40	2,155,953,248	0.3395	75.5712	24.7684
7.45	2,165,195,852	0.3410	75.2316	25.1093
7.50	2,196,777,968	0.3459	74.8907	25.4553
7.55	2,278,106,460	0.3587	74.5447	25.8140
7.60	2,343,902,648	0.3691	74.1860	26.1831
7.65	2,035,758,772	0.3206	73.8169	26.5037
7.70	2,379,814,608	0.3748	73.4963	26.8785
7.75	2,302,810,312	0.3626	73.1215	27.2411
7.80	2,290,664,488	0.3607	72.7589	27.6018
7.85	2,166,624,528	0.3412	72.3982	27.9430
7.90	2,383,072,632	0.3753	72.0570	28.3183
7.95	2,124,974,344	0.3346	71.6817	28.6529

Points	Hands	Specific	At Least	At Most
8	2,682,577,420	0.4224	71.3471	29.0754
8.05	2,042,136,768	0.3216	70.9246	29.3970
8.10	2,382,568,824	0.3752	70.6030	29.7722
8.15	2,411,064,768	0.3797	70.2278	30.1519
8.20	2,451,496,996	0.3861	69.8481	30.5379
8.25	2,216,633,496	0.3491	69.4621	30.8870
8.30	2,619,438,016	0.4125	69.1130	31.2995
8.35	2,300,090,004	0.3622	68.7005	31.6617
8.40	2,425,422,120	0.3819	68.3383	32.0437
8.45	2,464,654,800	0.3881	67.9563	32.4318
8.50	2,527,111,992	0.3980	67.5682	32.8297
8.55	2,295,431,636	0.3615	67.1703	33.1912
8.60	2,525,031,524	0.3976	66.8088	33.5889
8.65	2,584,002,364	0.4069	66.4111	33.9958
8.70	2,282,781,888	0.3595	66.0042	34.3553
8.75	2,547,730,392	0.4012	65.6447	34.7565
8.80	2,553,221,356	0.4021	65.2435	35.1585
8.85	2,344,043,768	0.3691	64.8415	35.5277
8.90	2,436,898,564	0.3838	64.4723	35.9114
8.95	2,554,011,236	0.4022	64.0886	36.3136

Points	Hands	Specific	At Least	At Most
9	2,506,481,684	0.3947	63.6864	36.7083
9.05	2,437,810,572	0.3839	63.2917	37.0922
9.10	2,574,684,544	0.4055	62.9078	37.4977
9.15	2,362,297,456	0.3720	62.5023	37.8697
9.20	2,701,100,396	0.4254	62.1303	38.2951
9.25	2,416,818,776	0.3806	61.7049	38.6757
9.30	2,583,979,656	0.4069	61.3243	39.0826
9.35	2,414,594,372	0.3802	60.9174	39.4628
9.40	2,753,816,332	0.4337	60.5372	39.8965
9.45	2,426,701,604	0.3821	60.1035	40.2786
9.50	2,726,477,332	0.4294	59.7214	40.7080
9.55	2,372,588,516	0.3736	59.2920	41.0816
9.60	2,786,591,492	0.4388	58.9184	41.5204
9.65	2,363,554,764	0.3722	58.4796	41.8926
9.70	2,671,708,236	0.4207	58.1074	42.3134
9.75	2,540,702,292	0.4001	57.6866	42.7135
9.80	2,723,284,500	0.4289	57.2865	43.1423
9.85	2,703,933,976	0.4258	56.8577	43.5681
9.90	2,606,786,156	0.4105	56.4319	43.9786
9.95	2,574,991,352	0.4055	56.0214	44.3841

Points	Hands	Specific	At Least	At Most
10	2,739,332,272	0.4314	55.6159	44.8155
10.05	2,655,382,052	0.4182	55.1845	45.2337
10.10	2,574,527,372	0.4054	54.7663	45.6391
10.15	2,660,718,204	0.4190	54.3609	46.0581
10.20	2,557,078,988	0.4027	53.9419	46.4608
10.25	2,634,119,140	0.4148	53.5392	46.8756
10.30	2,502,626,164	0.3941	53.1244	47.2697
10.35	2,497,402,568	0.3933	52.7303	47.6630
10.40	2,440,511,308	0.3843	52.3370	48.0473
10.45	2,564,844,620	0.4039	51.9527	48.4512
10.50	2,669,569,216	0.4204	51.5488	48.8716
10.55	2,411,248,764	0.3797	51.1284	49.2513
10.60	2,815,611,772	0.4434	50.7487	49.6947
10.65	2,323,263,460	0.3659	50.3053	50.0606
10.70	2,641,362,160	0.4160	49.9394	50.4766
10.75	2,511,085,348	0.3954	49.5234	50.8720
10.80	2,707,583,128	0.4264	49.1280	51.2984
10.85	2,316,428,844	0.3648	48.7016	51.6632
10.90	2,791,447,780	0.4396	48.3368	52.1027
10.95	2,253,079,136	0.3548	47.8973	52.4576

Points	Hands	Specific	At Least	At Most
11	2,806,684,160	0.4420	47.5424	52.8995
11.05	2,404,495,332	0.3787	47.1005	53.2782
11.10	2,506,036,592	0.3946	46.7218	53.6728
11.15	2,402,737,672	0.3784	46.3272	54.0512
11.20	2,817,112,924	0.4436	45.9488	54.4948
11.25	2,532,254,852	0.3988	45.5052	54.8936
11.30	2,446,405,572	0.3853	45.1064	55.2789
11.35	2,716,100,988	0.4277	44.7211	55.7066
11.40	2,558,501,064	0.4029	44.2934	56.1095
11.45	2,453,749,768	0.3864	43.8905	56.4959
11.50	2,653,624,632	0.4179	43.5041	56.9138
11.55	2,548,427,584	0.4013	43.0862	57.3151
11.60	2,398,493,732	0.3777	42.6849	57.6928
11.65	2,594,252,316	0.4085	42.3072	58.1014
11.70	2,423,502,268	0.3816	41.8986	58.4830
11.75	2,413,875,856	0.3801	41.5170	58.8631
11.80	2,499,958,800	0.3937	41.1369	59.2568
11.85	2,369,000,924	0.3731	40.7432	59.6299
11.90	2,448,854,700	0.3856	40.3701	60.0155
11.95	2,317,505,612	0.3650	39.9845	60.3805

Points	Hands	Specific	At Least	At Most
12	2,589,655,388	0.4078	39.6195	60.7883
12.05	2,287,816,220	0.3603	39.2117	61.1486
12.10	2,557,864,068	0.4028	38.8514	61.5514
12.15	2,197,616,516	0.3461	38.4486	61.8974
12.20	2,521,288,984	0.3970	38.1026	62.2945
12.25	2,226,369,104	0.3506	37.7055	62.6451
12.30	2,479,222,640	0.3904	37.3549	63.0355
12.35	2,260,753,740	0.3560	36.9645	63.3915
12.40	2,506,551,620	0.3947	36.6085	63.7862
12.45	2,141,708,728	0.3373	36.2138	64.1235
12.50	2,476,664,704	0.3900	35.8765	64.5135
12.55	2,270,739,044	0.3576	35.4865	64.8711
12.60	2,493,161,876	0.3926	35.1289	65.2637
12.65	2,251,130,924	0.3545	34.7363	65.6182
12.70	2,404,354,488	0.3786	34.3818	65.9969
12.75	2,345,708,068	0.3694	34.0031	66.3663
12.80	2,350,902,636	0.3702	33.6337	66.7365
12.85	2,364,630,736	0.3724	33.2635	67.1089
12.90	2,265,056,376	0.3567	32.8911	67.4655
12.95	2,182,202,468	0.3436	32.5345	67.8092

Points	Hands	Specific	At Least	At Most
13	2,411,457,368	0.3797	32.1908	68.1889
13.05	2,203,233,492	0.3470	31.8111	68.5359
13.10	2,188,709,520	0.3447	31.4641	68.8806
13.15	2,152,523,204	0.3390	31.1194	69.2195
13.20	2,192,382,276	0.3452	30.7805	69.5648
13.25	2,187,162,860	0.3444	30.4352	69.9092
13.30	2,221,897,128	0.3499	30.0908	70.2591
13.35	2,073,079,940	0.3265	29.7409	70.5856
13.40	2,221,708,764	0.3499	29.4144	70.9354
13.45	1,984,786,852	0.3126	29.0646	71.2480
13.50	2,231,559,828	0.3514	28.7520	71.5994
13.55	1,962,110,672	0.3090	28.4006	71.9084
13.60	2,181,195,500	0.3435	28.0916	72.2519
13.65	2,022,232,740	0.3185	27.7481	72.5704
13.70	2,076,140,176	0.3269	27.4296	72.8973
13.75	1,997,659,916	0.3146	27.1027	73.2119
13.80	2,157,356,392	0.3397	26.7881	73.5516
13.85	1,833,783,220	0.2888	26.4484	73.8404
13.90	2,146,992,436	0.3381	26.1596	74.1785
13.95	2,003,897,816	0.3156	25.8215	74.4941

Points	Hands	Specific	At Least	At Most
14	2,069,788,812	0.3259	25.5059	74.8200
14.05	1,930,222,552	0.3040	25.1800	75.1240
14.10	2,118,682,528	0.3336	24.8760	75.4576
14.15	1,857,456,780	0.2925	24.5424	75.7501
14.20	2,019,882,084	0.3181	24.2499	76.0682
14.25	1,972,481,616	0.3106	23.9318	76.3788
14.30	1,953,448,664	0.3076	23.6212	76.6865
14.35	1,832,249,620	0.2885	23.3135	76.9750
14.40	2,032,300,772	0.3200	23.0250	77.2950
14.45	1,825,964,644	0.2875	22.7050	77.5826
14.50	1,884,580,756	0.2968	22.4174	77.8794
14.55	1,782,774,532	0.2807	22.1206	78.1601
14.60	1,841,187,832	0.2899	21.8399	78.4500
14.65	1,767,828,744	0.2784	21.5500	78.7284
14.70	1,867,547,048	0.2941	21.2716	79.0225
14.75	1,665,704,040	0.2623	20.9775	79.2848
14.80	1,775,069,400	0.2795	20.7152	79.5644
14.85	1,699,318,340	0.2676	20.4356	79.8320
14.90	1,730,005,240	0.2724	20.1680	80.1044
14.95	1,591,301,776	0.2506	19.8956	80.3550

Points	Hands	Specific	At Least	At Most
15	1,814,224,944	0.2857	19.6450	80.6407
15.05	1,593,324,824	0.2509	19.3593	80.8916
15.10	1,739,708,644	0.2740	19.1084	81.1656
15.15	1,555,245,640	0.2449	18.8344	81.4105
15.20	1,714,106,728	0.2699	18.5895	81.6804
15.25	1,595,358,020	0.2512	18.3196	81.9317
15.30	1,720,194,052	0.2709	18.0683	82.2026
15.35	1,581,340,020	0.2490	17.7974	82.4516
15.40	1,654,765,492	0.2606	17.5484	82.7122
15.45	1,506,769,232	0.2373	17.2878	82.9495
15.50	1,678,710,672	0.2644	17.0505	83.2138
15.55	1,524,055,472	0.2400	16.7862	83.4538
15.60	1,643,567,024	0.2588	16.5462	83.7126
15.65	1,428,043,072	0.2249	16.2874	83.9375
15.70	1,547,280,072	0.2437	16.0625	84.1812
15.75	1,524,945,496	0.2401	15.8188	84.4213
15.80	1,513,151,024	0.2383	15.5787	84.6596
15.85	1,459,214,096	0.2298	15.3404	84.8894
15.90	1,473,311,956	0.2320	15.1106	85.1214
15.95	1,362,681,656	0.2146	14.8786	85.3360

Points	Hands	Specific	At Least	At Most
16	1,469,608,096	0.2314	14.6640	85.5674
16.05	1,360,018,632	0.2142	14.4326	85.7816
16.10	1,414,776,492	0.2228	14.2184	86.0044
16.15	1,324,492,028	0.2086	13.9956	86.2130
16.20	1,382,160,608	0.2177	13.7870	86.4306
16.25	1,308,314,840	0.2060	13.5694	86.6367
16.30	1,359,156,480	0.2140	13.3633	86.8507
16.35	1,246,390,728	0.1963	13.1493	87.0470
16.40	1,339,061,644	0.2109	12.9530	87.2579
16.45	1,175,265,868	0.1851	12.7421	87.4429
16.50	1,383,313,048	0.2178	12.5571	87.6608
16.55	1,174,061,908	0.1849	12.3392	87.8457
16.60	1,302,289,200	0.2051	12.1543	88.0507
16.65	1,152,464,632	0.1815	11.9493	88.2322
16.70	1,262,198,060	0.1988	11.7678	88.4310
16.75	1,137,870,312	0.1792	11.5690	88.6102
16.80	1,277,218,840	0.2011	11.3898	88.8113
16.85	1,120,273,716	0.1764	11.1887	88.9877
16.90	1,231,572,200	0.1939	11.0123	89.1817
16.95	1,083,928,852	0.1707	10.8183	89.3524

Points	Hands	Specific	At Least	At Most
17	1,221,136,548	0.1923	10.6476	89.5447
17.05	1,068,160,360	0.1682	10.4553	89.7129
17.10	1,148,548,100	0.1809	10.2871	89.8938
17.15	1,063,154,620	0.1674	10.1062	90.0612
17.20	1,131,178,196	0.1781	9.9388	90.2393
17.25	1,034,148,820	0.1629	9.7607	90.4022
17.30	1,088,012,436	0.1713	9.5978	90.5735
17.35	991,393,460	0.1561	9.4265	90.7296
17.40	1,072,218,320	0.1688	9.2704	90.8985
17.45	986,516,616	0.1554	9.1015	91.0538
17.50	1,021,442,548	0.1609	8.9462	91.2147
17.55	936,931,196	0.1475	8.7853	91.3622
17.60	1,063,506,136	0.1675	8.6378	91.5297
17.65	901,242,824	0.1419	8.4703	91.6716
17.70	957,009,236	0.1507	8.3284	91.8223
17.75	920,533,944	0.1450	8.1777	91.9673
17.80	946,088,580	0.1490	8.0327	92.1163
17.85	864,720,780	0.1362	7.8837	92.2525
17.90	975,912,472	0.1537	7.7475	92.4061
17.95	810,282,396	0.1276	7.5939	92.5337

Points	Hands	Specific	At Least	At Most
18	927,769,036	0.1461	7.4663	92.6798
18.05	799,792,648	0.1259	7.3202	92.8058
18.10	912,164,520	0.1436	7.1942	92.9494
18.15	793,569,584	0.1250	7.0506	93.0744
18.20	865,332,204	0.1363	6.9256	93.2107
18.25	789,779,640	0.1244	6.7893	93.3350
18.30	826,390,644	0.1301	6.6650	93.4652
18.35	757,788,276	0.1193	6.5348	93.5845
18.40	804,567,428	0.1267	6.4155	93.7112
18.45	714,221,372	0.1125	6.2888	93.8237
18.50	804,705,560	0.1267	6.1763	93.9504
18.55	717,462,408	0.1130	6.0496	94.0634
18.60	760,722,668	0.1198	5.9366	94.1832
18.65	690,050,060	0.1087	5.8168	94.2919
18.70	706,304,324	0.1112	5.7081	94.4031
18.75	682,645,528	0.1075	5.5969	94.5106
18.80	745,078,944	0.1173	5.4894	94.6279
18.85	617,550,356	0.0972	5.3721	94.7252
18.90	692,538,868	0.1091	5.2748	94.8342
18.95	620,896,084	0.0978	5.1658	94.9320

Points	Hands	Specific	At Least	At Most
19	684,502,240	0.1078	5.0680	95.0398
19.05	611,759,724	0.0963	4.9602	95.1361
19.10	643,944,516	0.1014	4.8639	95.2375
19.15	580,434,240	0.0914	4.7625	95.3290
19.20	642,247,308	0.1011	4.6710	95.4301
19.25	575,866,228	0.0907	4.5699	95.5208
19.30	624,123,880	0.0983	4.4792	95.6191
19.35	531,126,396	0.0836	4.3809	95.7027
19.40	614,316,352	0.0967	4.2973	95.7994
19.45	525,218,200	0.0827	4.2006	95.8822
19.50	605,372,720	0.0953	4.1178	95.9775
19.55	520,194,604	0.0819	4.0225	96.0594
19.60	556,740,452	0.0877	3.9406	96.1471
19.65	502,107,964	0.0791	3.8529	96.2261
19.70	552,542,060	0.0870	3.7739	96.3132
19.75	470,776,052	0.0741	3.6868	96.3873
19.80	531,970,000	0.0838	3.6127	96.4711
19.85	458,214,428	0.0722	3.5289	96.5432
19.90	496,023,956	0.0781	3.4568	96.6213
19.95	444,272,392	0.0700	3.3787	96.6913

Points	Hands	Specific	At Least	At Most
20	488,506,028	0.0769	3.3087	96.7682
20.05	426,333,088	0.0671	3.2318	96.8354
20.10	474,974,692	0.0748	3.1646	96.9102
20.15	419,486,588	0.0661	3.0898	96.9762
20.20	455,794,528	0.0718	3.0238	97.0480
20.25	388,332,216	0.0612	2.9520	97.1092
20.30	446,237,432	0.0703	2.8908	97.1794
20.35	394,165,340	0.0621	2.8206	97.2415
20.40	414,112,692	0.0652	2.7585	97.3067
20.45	362,960,628	0.0572	2.6933	97.3639
20.50	403,234,076	0.0635	2.6361	97.4274
20.55	350,654,624	0.0552	2.5726	97.4826
20.60	397,575,176	0.0626	2.5174	97.5452
20.65	341,377,484	0.0538	2.4548	97.5990
20.70	367,480,556	0.0579	2.4010	97.6568
20.75	327,784,192	0.0516	2.3432	97.7084
20.80	377,942,608	0.0595	2.2916	97.7680
20.85	316,514,784	0.0498	2.2320	97.8178
20.90	342,357,080	0.0539	2.1822	97.8717
20.95	299,432,620	0.0472	2.1283	97.9189

Points	Hands	Specific	At Least	At Most
21	348,052,380	0.0548	2.0811	97.9737
21.05	283,806,816	0.0447	2.0263	98.0184
21.10	338,242,680	0.0533	1.9816	98.0716
21.15	266,519,536	0.0420	1.9284	98.1136
21.20	321,234,952	0.0506	1.8864	98.1642
21.25	265,842,256	0.0419	1.8358	98.2061
21.30	297,160,504	0.0468	1.7939	98.2529
21.35	254,382,252	0.0401	1.7471	98.2929
21.40	303,215,760	0.0477	1.7071	98.3407
21.45	232,017,336	0.0365	1.6593	98.3772
21.50	273,802,720	0.0431	1.6228	98.4203
21.55	233,883,504	0.0368	1.5797	98.4572
21.60	265,521,816	0.0418	1.5428	98.4990
21.65	231,071,648	0.0364	1.5010	98.5354
21.70	249,310,992	0.0393	1.4646	98.5746
21.75	213,197,844	0.0336	1.4254	98.6082
21.80	255,197,712	0.0402	1.3918	98.6484
21.85	206,124,320	0.0325	1.3516	98.6808
21.90	226,879,028	0.0357	1.3192	98.7166
21.95	202,827,520	0.0319	1.2834	98.7485

Points	Hands	Specific	At Least	At Most
22	224,370,816	0.0353	1.2515	98.7838
22.05	190,848,440	0.0301	1.2162	98.8139
22.10	215,306,936	0.0339	1.1861	98.8478
22.15	181,037,120	0.0285	1.1522	98.8763
22.20	199,057,956	0.0313	1.1237	98.9077
22.25	179,244,948	0.0282	1.0923	98.9359
22.30	191,144,704	0.0301	1.0641	98.9660
22.35	160,099,924	0.0252	1.0340	98.9912
22.40	190,549,780	0.0300	1.0088	99.0212
22.45	160,849,720	0.0253	0.9788	99.0465
22.50	171,944,412	0.0271	0.9535	99.0736
22.55	146,371,700	0.0231	0.9264	99.0967
22.60	175,518,100	0.0276	0.9033	99.1243
22.65	137,548,484	0.0217	0.8757	99.1460
22.70	164,890,300	0.0260	0.8540	99.1719
22.75	132,869,256	0.0209	0.8281	99.1929
22.80	154,515,168	0.0243	0.8071	99.2172
22.85	127,254,336	0.0200	0.7828	99.2372
22.90	143,767,764	0.0226	0.7628	99.2599
22.95	118,942,780	0.0187	0.7401	99.2786

Points	Hands	Specific	At Least	At Most
23	151,260,736	0.0238	0.7214	99.3024
23.05	112,706,408	0.0177	0.6976	99.3202
23.10	134,904,452	0.0212	0.6798	99.3414
23.15	111,675,780	0.0176	0.6586	99.3590
23.20	129,522,888	0.0204	0.6410	99.3794
23.25	104,966,116	0.0165	0.6206	99.3959
23.30	120,934,960	0.0190	0.6041	99.4150
23.35	101,568,508	0.0160	0.5850	99.4310
23.40	117,170,204	0.0185	0.5690	99.4494
23.45	88,748,120	0.0140	0.5506	99.4634
23.50	110,770,116	0.0174	0.5366	99.4808
23.55	89,708,920	0.0141	0.5192	99.4950
23.60	106,116,888	0.0167	0.5050	99.5117
23.65	86,438,996	0.0136	0.4883	99.5253
23.70	99,129,280	0.0156	0.4747	99.5409
23.75	79,511,520	0.0125	0.4591	99.5534
23.80	97,853,704	0.0154	0.4466	99.5688
23.85	81,282,084	0.0128	0.4312	99.5816
23.90	90,019,932	0.0142	0.4184	99.5958
23.95	69,061,260	0.0109	0.4042	99.6067

Points	Hands	Specific	At Least	At Most
24	89,930,228	0.0142	0.3933	99.6208
24.05	67,220,940	0.0106	0.3792	99.6314
24.10	78,122,288	0.0123	0.3686	99.6437
24.15	65,541,852	0.0103	0.3563	99.6540
24.20	74,442,996	0.0117	0.3460	99.6658
24.25	60,475,736	0.0095	0.3342	99.6753
24.30	73,401,616	0.0116	0.3247	99.6869
24.35	55,797,312	0.0088	0.3131	99.6956
24.40	70,318,452	0.0111	0.3044	99.7067
24.45	53,768,040	0.0085	0.2933	99.7152
24.50	64,298,936	0.0101	0.2848	99.7253
24.55	52,121,104	0.0082	0.2747	99.7335
24.60	65,011,612	0.0102	0.2665	99.7438
24.65	47,514,068	0.0075	0.2562	99.7512
24.70	58,759,172	0.0093	0.2488	99.7605
24.75	43,274,172	0.0068	0.2395	99.7673
24.80	57,189,084	0.0090	0.2327	99.7763
24.85	41,217,136	0.0065	0.2237	99.7828
24.90	52,279,360	0.0082	0.2172	99.7910
24.95	38,316,552	0.0060	0.2090	99.7971

Points	Hands	Specific	At Least	At Most
25	49,002,264	0.0077	0.2029	99.8048
25.05	39,923,060	0.0063	0.1952	99.8111
25.10	43,238,420	0.0068	0.1889	99.8179
25.15	35,790,440	0.0056	0.1821	99.8235
25.20	43,285,544	0.0068	0.1765	99.8303
25.25	30,424,092	0.0048	0.1697	99.8351
25.30	39,945,036	0.0063	0.1649	99.8414
25.35	31,322,512	0.0049	0.1586	99.8463
25.40	37,606,212	0.0059	0.1537	99.8523
25.45	29,942,428	0.0047	0.1477	99.8570
25.50	35,485,528	0.0056	0.1430	99.8626
25.55	27,069,072	0.0043	0.1374	99.8668
25.60	35,967,948	0.0057	0.1332	99.8725
25.65	25,309,820	0.0040	0.1275	99.8765
25.70	32,393,256	0.0051	0.1235	99.8816
25.75	22,854,664	0.0036	0.1184	99.8852
25.80	30,100,500	0.0047	0.1148	99.8899
25.85	23,010,160	0.0036	0.1101	99.8935
25.90	29,816,932	0.0047	0.1065	99.8982
25.95	20,620,140	0.0032	0.1018	99.9015

Points	Hands	Specific	At Least	At Most
26	26,380,440	0.0042	0.0985	99.9056
26.05	19,204,996	0.0030	0.0944	99.9087
26.10	24,902,448	0.0039	0.0913	99.9126
26.15	17,346,016	0.0027	0.0874	99.9153
26.20	24,397,380	0.0038	0.0847	99.9192
26.25	17,311,060	0.0027	0.0808	99.9219
26.30	21,286,052	0.0034	0.0781	99.9252
26.35	15,933,492	0.0025	0.0748	99.9278
26.40	21,991,224	0.0035	0.0722	99.9312
26.45	14,804,452	0.0023	0.0688	99.9335
26.50	18,774,108	0.0030	0.0665	99.9365
26.55	13,837,432	0.0022	0.0635	99.9387
26.60	17,655,984	0.0028	0.0613	99.9415
26.65	13,056,460	0.0021	0.0585	99.9435
26.70	15,700,772	0.0025	0.0565	99.9460
26.75	11,992,524	0.0019	0.0540	99.9479
26.80	14,603,116	0.0023	0.0521	99.9502
26.85	11,336,572	0.0018	0.0498	99.9520
26.90	13,798,400	0.0022	0.0480	99.9541
26.95	9,969,896	0.0016	0.0459	99.9557

Points	Hands	Specific	At Least	At Most
27	11,962,932	0.0019	0.0443	99.9576
27.05	9,764,956	0.0015	0.0424	99.9591
27.10	12,514,240	0.0020	0.0409	99.9611
27.15	8,192,196	0.0013	0.0389	99.9624
27.20	12,434,364	0.0020	0.0376	99.9643
27.25	7,070,156	0.0011	0.0357	99.9655
27.30	10,863,564	0.0017	0.0345	99.9672
27.35	6,631,852	0.0010	0.0328	99.9682
27.40	10,788,412	0.0017	0.0318	99.9699
27.45	6,480,088	0.0010	0.0301	99.9709
27.50	9,704,884	0.0015	0.0291	99.9725
27.55	6,029,996	0.0009	0.0275	99.9734
27.60	8,575,260	0.0014	0.0266	99.9748
27.65	5,723,304	0.0009	0.0252	99.9757
27.70	7,962,860	0.0013	0.0243	99.9769
27.75	5,350,392	0.0008	0.0231	99.9778
27.80	7,450,648	0.0012	0.0222	99.9789
27.85	5,159,288	0.0008	0.0211	99.9797
27.90	6,254,972	0.0010	0.0203	99.9807
27.95	4,898,404	0.0008	0.0193	99.9815

Points	Hands	Specific	At Least	At Most
28	7,098,656	0.0011	0.0185	99.9826
28.05	4,282,456	0.0007	0.0174	99.9833
28.10	5,638,652	0.0009	0.0167	99.9842
28.15	3,708,532	0.0006	0.0158	99.9848
28.20	5,769,760	0.0009	0.0152	99.9857
28.25	3,569,456	0.0006	0.0143	99.9862
28.30	5,018,140	0.0008	0.0138	99.9870
28.35	3,407,176	0.0005	0.0130	99.9876
28.40	4,257,924	0.0007	0.0124	99.9882
28.45	3,007,112	0.0005	0.0118	99.9887
28.50	4,071,820	0.0006	0.0113	99.9894
28.55	2,524,688	0.0004	0.0106	99.9897
28.60	3,610,104	0.0006	0.0103	99.9903
28.65	2,225,308	0.0004	0.0097	99.9907
28.70	3,590,100	0.0006	0.0093	99.9912
28.75	1,854,464	0.0003	0.0088	99.9915
28.80	2,922,012	0.0005	0.0085	99.9920
28.85	2,151,280	0.0003	0.0080	99.9923
28.90	2,984,276	0.0005	0.0077	99.9928
28.95	1,732,564	0.0003	0.0072	99.9931

Points	Hands	Specific	At Least	At Most
29	2,873,116	0.0005	0.0069	99.9935
29.05	1,497,800	0.0002	0.0065	99.9938
29.10	2,400,568	0.0004	0.0062	99.9941
29.15	1,573,884	0.0002	0.0059	99.9944
29.20	2,213,592	0.0003	0.0056	99.9947
29.25	1,275,308	0.0002	0.0053	99.9949
29.30	2,118,844	0.0003	0.0051	99.9953
29.35	1,254,132	0.0002	0.0047	99.9955
29.40	2,065,212	0.0003	0.0045	99.9958
29.45	1,192,304	0.0002	0.0042	99.9960
29.50	1,494,948	0.0002	0.0040	99.9962
29.55	1,017,456	0.0002	0.0038	99.9964
29.60	1,501,368	0.0002	0.0036	99.9966
29.65	1,050,364	0.0002	0.0034	99.9968
29.70	1,336,752	0.0002	0.0032	99.9970
29.75	920,552	0.0001	0.0030	99.9971
29.80	1,303,104	0.0002	0.0029	99.9973
29.85	622,036	0.0001	0.0027	99.9974
29.90	1,231,704	0.0002	0.0026	99.9976
29.95	590,672	0.0001	0.0024	99.9977

Points	Hands	Specific	At Least	At Most
30	1,059,784	0.0002	0.0023	99.9979
30.05	582,408	0.0001	0.0021	99.9980
30.10	888,924	0.0001	0.0020	99.9981
30.15	526,688	0.0001	0.0019	99.9982
30.20	900,588	0.0001	0.0018	99.9983
30.25	423,304	0.0001	0.0017	99.9984
30.30	764,236	0.0001	0.0016	99.9985
30.35	416,588	0.0001	0.0015	99.9986
30.40	631,452	0.0001	0.0014	99.9987
30.45	372,360	0.0001	0.0013	99.9988
30.50	532,572	0.0001	0.0012	99.9988
30.55	382,836	0.0001	0.0012	99.9989
30.60	609,148	0.0001	0.0011	99.9990
30.65	273,000	0.0000	0.0010	99.9990
30.70	477,052	0.0001	0.0010	99.9991
30.75	201,132	0.0000	0.0009	99.9991
30.80	503,640	0.0001	0.0009	99.9992
30.85	232,712	0.0000	0.0008	99.9993
30.90	466,808	0.0001	0.0007	99.9993
30.95	220,276	0.0000	0.0007	99.9994

Points	Hands	Specific	At Least	At Most
31	243,700	0.0000	0.0006	99.9994
31.05	181,888	0.0000	0.0006	99.9994
31.10	327,676	0.0001	0.0006	99.9995
31.15	178,744	0.0000	0.0005	99.9995
31.20	246,908	0.0000	0.0005	99.9996
31.25	158,400	0.0000	0.0004	99.9996
31.30	250,020	0.0000	0.0004	99.9996
31.35	78,544	0.0000	0.0004	99.9996
31.40	273,132	0.0000	0.0004	99.9997
31.45	94,288	0.0000	0.0003	99.9997
31.50	122,720	0.0000	0.0003	99.9997
31.55	93,460	0.0000	0.0003	99.9997
31.60	183,180	0.0000	0.0003	99.9997
31.65	60,856	0.0000	0.0003	99.9998
31.70	145,836	0.0000	0.0002	99.9998
31.75	81,168	0.0000	0.0002	99.9998
31.80	134,896	0.0000	0.0002	99.9998
31.85	45,292	0.0000	0.0002	99.9998
31.90	127,916	0.0000	0.0002	99.9998
31.95	49,764	0.0000	0.0002	99.9999

Points	Hands	Specific	At Least	At Most
32	94,824	0.0000	0.0001	99.9999
32.05	43,408	0.0000	0.0001	99.9999
32.10	72,016	0.0000	0.0001	99.9999
32.15	35,328	0.0000	0.0001	99.9999
32.20	82,000	0.0000	0.0001	99.9999
32.25	44,236	0.0000	0.0001	99.9999
32.30	72,484	0.0000	0.0001	99.9999
32.35	29,940	0.0000	0.0001	99.9999
32.40	63,352	0.0000	0.0001	99.9999
32.45	19,856	0.0000	0.0001	99.9999
32.50	43,596	0.0000	0.0001	99.9999
32.55	23,496	0.0000	0.0001	99.9999
32.60	27,588	0.0000	0.0001	100.0000
32.65	13,732	0.0000	0.0000	100.0000
32.70	36,576	0.0000	0.0000	100.0000
32.75	20,148	0.0000	0.0000	100.0000
32.80	41,908	0.0000	0.0000	100.0000
32.85	7432	0.0000	0.0000	100.0000
32.90	20,752	0.0000	0.0000	100.0000
32.95	5268	0.0000	0.0000	100.0000

Points	Hands	Specific	At Least	At Most
33	11,348	0.0000	0.0000	100.0000
33.05	10,476	0.0000	0.0000	100.0000
33.10	14,776	0.0000	0.0000	100.0000
33.15	5364	0.0000	0.0000	100.0000
33.20	14,472	0.0000	0.0000	100.0000
33.25	3828	0.0000	0.0000	100.0000
33.30	13,036	0.0000	0.0000	100.0000
33.35	1760	0.0000	0.0000	100.0000
33.40	13,668	0.0000	0.0000	100.0000
33.45	2592	0.0000	0.0000	100.0000
33.50	8400	0.0000	0.0000	100.0000
33.55	3804	0.0000	0.0000	100.0000
33.60	6496	0.0000	0.0000	100.0000
33.65	936	0.0000	0.0000	100.0000
33.70	5316	0.0000	0.0000	100.0000
33.75	972	0.0000	0.0000	100.0000
33.80	7648	0.0000	0.0000	100.0000
33.85	2436	0.0000	0.0000	100.0000
33.90	3384	0.0000	0.0000	100.0000
33.95	228	0.0000	0.0000	100.0000

Points	Hands	Specific	At Least	At Most
34	2364	0.0000	0.0000	100.0000
34.05	724	0.0000	0.0000	100.0000
34.10	1500	0.0000	0.0000	100.0000
34.15	732	0.0000	0.0000	100.0000
34.20	1212	0.0000	0.0000	100.0000
34.25	52	0.0000	0.0000	100.0000
34.30	2044	0.0000	0.0000	100.0000
34.35	156	0.0000	0.0000	100.0000
34.40	1832	0.0000	0.0000	100.0000
34.50	796	0.0000	0.0000	100.0000
34.55	636	0.0000	0.0000	100.0000
34.60	252	0.0000	0.0000	100.0000
34.65	84	0.0000	0.0000	100.0000
34.70	412	0.0000	0.0000	100.0000
34.80	672	0.0000	0.0000	100.0000
34.85	12	0.0000	0.0000	100.0000
34.90	132	0.0000	0.0000	100.0000
35.05	36	0.0000	0.0000	100.0000
35.10	144	0.0000	0.0000	100.0000
35.20	28	0.0000	0.0000	100.0000
35.30	36	0.0000	0.0000	100.0000
35.40	16	0.0000	0.0000	100.0000
35.60	8	0.0000	0.0000	100

Totals (35 tables)

Unique Values	Hands	Specific	At Least + At Most
710 = U	635,013,559,600	100	71100*

*Check total should be exactly 100(U+1).

Study 8J19 Main

Top Kaplan-Rubens Evaluation Stats

© 2014 Richard Pavlicek