Dealt Badugi Probabilities

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[edit] Four-card Hands

What is the probability of being dealt a pat Badugi? What is the median Badugi?

We can calculate the number of badugis with high card N as ((N-1) choose 3) * 4!. (That is, we must choose 1 card of rank N, 3 different ranks from the remaining N-1 ranks, and they may have any assignment of four different suits.)

Rank	Number of Hands	Absolute Probability	Relative Probability Within Badugis
432A	24	0.00008865	0.00139860
5-high	96	0.00035460	0.00559441
6-high	240	0.00088651	0.01398601
7-high	480	0.00177302	0.02797203
8-high	840	0.00310278	0.04895105
9-high	1344	0.00496445	0.07832168
10-high	2016	0.00744667	0.11748252
J-high	2880	0.01063810	0.16783217
Q-high	3960	0.01462739	0.23076923
K-high	5280	0.01950319	0.30769231
Total	17160	0.06338535	1.00000000

The median badugi is number 8580, which places it 660 hands into the Q's.

Rank	Number of Hands
Q32A	24
Q4xx	72
Q5xx	144
Q6xx	240
Q72A	24
Q73x	48
Q74x	72
Q75A	24
Q752	24
Total	672

So, the median dealt pat badugi is Q752.

[edit] Three-card Hands

Example calculation for 42A:

There are 9 cards of the same suit as 4 but worse rank. There are 11 cards of the same suit as the deuce, with worse rank, but the '4' pairs, so count 10 unpaired cards. Similarly, there are 10 unpaired cards of worse rank than the 'A' in its suit. This gives (9+10+10)*4*3*2 = 696 unpaired 3-card hands.

For whatever rank of cards, there are 180 paired 3-card hands. Consider 42AA: there are four ways to pick the suit of the '4', three ways to pick the suit of the '3', and then five ways to pick the suit of the remaining pair, one of which must be a third suit in order to make a 3-card hand. This gives 60 combinations per pairing and three possible pairings, or 180 combinations.

The total is thus 876 ways to make a 42A which is not also a badugi.

Rank	Number of Hands	Absolute Probability	Relative Probability Within 3-Card Hands
32A	900	0.00332441	0.00582751
42A	876	0.00323576	0.00567211
43A	852	0.00314710	0.00551671
432	828	0.00305845	0.00536131
52A	852	0.00314710	0.00551671
53A	828	0.00305845	0.00536131
532	804	0.00296980	0.00520591
54A	804	0.00296980	0.00520591
542	780	0.00288115	0.00505051
543	756	0.00279250	0.00489510
62A	828	0.00305845	0.00536131
63A	804	0.00296980	0.00520591
632	780	0.00288115	0.00505051
64A	780	0.00288115	0.00505051
642	756	0.00279250	0.00489510
643	732	0.00270385	0.00473970
65A	756	0.00279250	0.00489510
652	732	0.00270385	0.00473970
653	708	0.00261520	0.00458430
654	684	0.00252655	0.00442890
72A	804	0.00296980	0.00520591
73A	780	0.00288115	0.00505051
732	756	0.00279250	0.00489510
74A	756	0.00279250	0.00489510
742	732	0.00270385	0.00473970
743	708	0.00261520	0.00458430
75A	732	0.00270385	0.00473970
752	708	0.00261520	0.00458430
753	684	0.00252655	0.00442890
754	660	0.00243790	0.00427350
76A	708	0.00261520	0.00458430
762	684	0.00252655	0.00442890
763	660	0.00243790	0.00427350
764	636	0.00234925	0.00411810
765	612	0.00226060	0.00396270
82A	780	0.00288115	0.00505051
83A	756	0.00279250	0.00489510
832	732	0.00270385	0.00473970
84A	732	0.00270385	0.00473970
842	708	0.00261520	0.00458430
843	684	0.00252655	0.00442890
85A	708	0.00261520	0.00458430
852	684	0.00252655	0.00442890
853	660	0.00243790	0.00427350
854	636	0.00234925	0.00411810
86A	684	0.00252655	0.00442890
862	660	0.00243790	0.00427350
863	636	0.00234925	0.00411810
864	612	0.00226060	0.00396270
865	588	0.00217195	0.00380730
87A	660	0.00243790	0.00427350
872	636	0.00234925	0.00411810
873	612	0.00226060	0.00396270
874	588	0.00217195	0.00380730
875	564	0.00208329	0.00365190
876	540	0.00199464	0.00349650
8xx	13860
9xx	17136
Txx	20304
Jxx	23220
Qxx	25740
Kxx	27720
Total	154440	0.56046819	1.00000000