Heads Up Hold'em Percentages

Posted to Rec.Gambling.Poker, 5 Feb 95 16:34:12
Copyright 1995,1998, Jazbo Enterprises

Has anyone calculated the chance that any particular 2-card hand will beat any randomly-dealt 2-card hand for all the possible random boards? I'd like to know for judging the relative value of 2-card hands.

There are 47,008 possible distinct head-to-head matchups. I have calculated the exact W/L/T values for all these (and boy are my arms tired).

From this, it isn't hard to calculate the expectation of any two card hand. The exact ranking of all 169 hands is given below. It is interesting to compare these values with Justin Cases' table in _Percentage Hold'Em_, which used simulation to estimate the hand values. I think the main difference is that I counted ties as worth 1/2, while I think Justin's table counted ties as 1.

Note: These numbers do not give necessarily indicate how hands match up against each other, but how each hand will do independently against a random, unknown hand. For example, Q7s (0.5430) ranks just above K6off (0.5422) in the table. That means, if you all-in preflop against an unknown hand, you would pick Q7s instead of K6off. However, if you have a proposition bet where you can take either Qd7d or Kh6c against each other, then you should take Kh6c, since it is favored against Qd7d.

AA0.8520371 A4s0.5903364 K50.5331397 96s0.4742829 850.4142753
KK0.8239568 A70.5884120 J90.5325120 J2s0.4737815 64s0.4133332
QQ0.7992516 K8s0.5831235 K2s0.5321173 Q20.4729544 83s0.4087350
JJ0.7746947 A3s0.5822032 Q5s0.5276941 T5s0.4721626 940.4067105
TT0.7501178 QJ0.5813469 T8s0.5233437 J50.4718089 750.4051197
990.7205725 K90.5781192 K40.5232747 T4s0.4653049 82s0.4027163
880.6916304 A50.5769653 J7s0.5232478 970.4629781 73s0.4003594
AKs0.6704463 A60.5768245 Q4s0.5185530 86s0.4624327 930.4001951
770.6623602 Q9s0.5766432 Q70.5176567 J40.4618638 650.3994430
AQs0.6620886 K7s0.5753774 T90.5153167 T60.4609200 53s0.3969296
AJs0.6539268 JTs0.5752786 J80.5149016 95s0.4572187 63s0.3953356
AK0.6532007 A2s0.5737890 K30.5142569 T3s0.4569251 840.3944679
ATs0.6460239 QT0.5729078 Q60.5102405 76s0.4537177 920.3909794
AQ0.6443184 440.5702282 Q3s0.5101925 J30.4527554 43s0.3864195
AJ0.6356326 A40.5672968 98s0.5080076 870.4505081 740.3854983
KQs0.6340040 K6s0.5664074 T7s0.5063904 T2s0.4483948 72s0.3815589
660.6328475 K80.5602017 J6s0.5060591 85s0.4454499 540.3815529
A9s0.6278121 Q8s0.5601773 K20.5050872 960.4449135 640.3801049
AT0.6272165 A30.5584460 220.5033402 J20.4434847 52s0.3784933
KJs0.6256734 K5s0.5579292 Q2s0.5016904 T50.4425095 62s0.3766896
A8s0.6194381 J9s0.5566247 Q50.5012008 94s0.4386197 830.3748381
KTs0.6178856 Q90.5536043 J5s0.4998685 75s0.4367554 42s0.3682901
KQ0.6145580 JT0.5524770 T80.4972127 T40.4350411 820.3682767
A7s0.6098396 K70.5518735 J70.4968193 93s0.4326426 730.3660226
A90.6077281 A20.5492856 Q40.4912768 860.4324090 530.3626477
KJ0.6056869 K4s0.5488464 97s0.4911773 65s0.4313339 630.3607763
550.6032492 Q7s0.5430226 J4s0.4907045 84s0.4270163 32s0.3598443
QJs0.6025921 K60.5422328 T6s0.4894068 950.4266914 430.3514589
K9s0.5998848 K3s0.5405498 J3s0.4823162 T30.4259455 720.3458365
A5s0.5992293 T9s0.5402753 Q30.4821944 92s0.4241517 520.3428465
A6s0.5990583 J8s0.5401564 980.4809703 760.4232275 620.3407514
A80.5987261 330.5369308 87s0.4793634 74s0.4184931 420.3319975
KT0.5973892 Q6s0.5361257 T70.4790814 T20.4166835 320.3230323
QTs0.5946756 Q80.5359979 J60.4784427 54s0.4145342

--jazbo