One of the best methods of
evaluating the strength of a hand is by adding up the point count. This
method awards 4 points for every Ace in a hand, 3 points for each King,
2 points for each Queen, and 1 point for each Jack. Additional points
are awarded for distributional strength such as a short suit, but here
we will just give the probabilities for raw honor count power.
The highest possible honor count that can exist in a hand
would have all four Aces, four Kings, four Queens, and one of the four
Jacks for a total of 37 points. At the other end of the scale there are
over 2 billion hands that have a zero point count (10 high or worse).
Of these, COMBIN(32, 13) = 347,373,600 are Yarboroughs (9 high or
less). We also note there are COMBIN(52, 13) = 635,013,599,600
different hands that could be dealt.
For each row in the table below, column 1 gives the point
count via the 4, 3, 2, 1 analysis. Note that there are many
combinations that can produce a given count. For example a hand that
has 2 Aces and a Queen, or another hand that has 4 Queens and 2 Jacks
would both be included in the "10" row.
The second column shows the exact number of possible hands
that will produce the given point count (Honors only - distribution is
not counted). For the third column we divide the total combinations in
the second column by the total number of all hands (635,013,559,600) to
get the probability of being dealt this particular point count. The
fourth column gives the cumulative probability of receiving a
particular point count or higher. Finally the fifth column shows the
average number of honor cards that a hand will have, given that the
hand has a particular honor count.
High point count hands have a very low probability, and
hence scientific notation is used. For example, to express the
probability of getting a 37 point hand as a fixed point decimal number,
you have to move the decimal point 12 places further to the left (e.g.
.00000000000629908)
Honor
Total Honor
Count
Cumulative Avg. Nbr.
Count
Hands
Probability
Probability of Honors
-----------------------------------------------------------------------
37
4 6.29908
E-12 6.29908 E-12
13.0000
36
60 9.44862
E-11 1.00785 E-10
12.4000
35
624 9.82656
E-10 1.08344 E-09
12.0769
34
4,484 7.06127
E-09 8.14471 E-09
11.4585
33
22,360 3.52118
E-08 4.33566 E-08
11.2161
32
109,156 1.71896
E-07 2.15252 E-07
10.6851
31
388,196 6.11319
E-07 8.26571 E-07
10.4401
30
1,396,068 2.19849
E-06 3.02506 E-06
10.0376
29
4,236,588 6.67165
E-06 9.69671 E-06
9.7116
28
11,790,760 1.85677
E-05 2.82644 E-05
9.4187
27
31,157,940 4.90666
E-05 7.73310 E-05
9.0614
26
74,095,248
0.000116683
0.000194014 8.7857
25
167,819,892
0.000264278
0.000458292 8.4670
24
354,993,864
0.000559034
0.00101733 8.1655
23
710,603,628
0.00111904
0.00213636 7.8697
22
1,333,800,036
0.00210043
0.00423679 7.5769
21
2,399,507,844
0.00377867
0.00801546 7.2797
20
4,086,538,404
0.00643536
0.0144508 6.9817
19
6,579,838,440
0.0103617
0.0248125 6.7023
18
10,192,504,020
0.0160508
0.0408634 6.3982
17
14,997,082,848
0.0236169
0.0644803 6.1113
16
21,024,781,756
0.0331092
0.0975895 5.8196
15
28,090,962,724
0.0442368
0.141826 5.5275
14
36,153,374,224
0.0569332
0.198760 5.2273
13
43,906,944,752
0.0691433
0.267903 4.9381
12
50,971,682,080
0.0802687
0.348172 4.6450
11
56,799,933,520
0.0894468
0.437618 4.3279
10
59,723,754,816
0.0940511
0.531669 4.0415
9 59,413,313,872
0.0935623
0.625232 3.7356
8 56,466,608,128
0.0889219
0.714154 3.4192
7 50,979,441,968
0.0802809
0.794435 3.0811
6 41,619,399,184
0.0655410
0.859975 2.8059
5 32,933,031,040
0.0518619
0.911837 2.4620
4 24,419,055,136
0.0384544
0.950292 2.0525
3 15,636,342,960
0.0246236
0.974915 1.7448
2
8,611,542,576
0.0135612
0.988477 1.4186
1
5,006,710,800
0.00788442
0.996361 1.0000
0
2,310,789,600
0.00363896
1.000000 0.0000
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