odds that my hand is dominated

[windmill]

Does anybody know where I can find odds that a specific hand is dominated (a couple of examples: you have A6 and one of the other players has Ax with x > 6, or another player has a pocket pair 6s or higher, or you have a pocket pair, but somebody else has a higher pocket pair). I would like to know the odds of having a dominated hand depending on the number of players at the table.

I hope somebody can help me out with this.

[man]

This is the most complicated question I have ever seen. If you want to mess around, you can use the odds calculator at cardplayer:

http://www.cardplayer.com/poker_odds/texas_holdem/index.php

Or you can look at the odds that certain hands have against other random hands at wizardofodds:

http://wizardofodds.com/holdem

But trying to understand how one would even begin to answer this question gives me a headache.

[windmill]

[ QUOTE ]
This is the most complicated question I have ever seen.

[/ QUOTE ]

Hmmm, yes, I'm starting to think this is indeed really complicated. Exact calculations are probably out of the question so I suppose I need to try to run some simulations.

[closer2313]

I dont think its that complicated. Here is an example if you are first in with AQo
Hands that dominate you
AK
AA
KK
QQ
KK - 6 combinations, AK- 12 combinations, AA - 3 QQ - 3
This is because you have an A and a Q in your hand.
So thats 24 combinations of hands that dominate you, out of C(50,2) = 1225.
Now it depends on how many people have to act behind you.. say you are first position at a 9 handed table. it would be
8 * 24/1225 ~15.6% of the time you will be dominated. But if you are opening from the button its 2 * 24 /1225 ~ 3.9%

[man]

you're right, it's not so bad to work out specific hand domination.

btw, your calculation is off. you can't simply add up the probability that you're dominated. you can find out the probability that one person is dominating you by finding the probability that you're not dominated by anyone and subtracting from one, though:

1 - (1201/1225)^8 = 14.64%

and similarly for the heads-up situation. to find if more than one person is dominating you is a bit more complex, but not too bad.

[closer2313]

My mistake, I think I wanted to start to do inclusion-exclusion and forgot to finish.

[Marc B]

I did these using simulations over about 10 million hands, so while the numbers aren't exact, they come close. In order from least likely to be dominated with (n-1) players remaining to most likely. Note that AKs == AK for the sake of this problem.

Ten handed
AA 0.007381046526954
AK 0.043748142767524
KK 0.0496223432285263
QQ 0.0893896630799083
JJ 0.13575304688195
AQ 0.165675966553523
TT 0.174328910829991
99 0.207368188837415
88 0.241813152493977
KQ 0.243355000909707
77 0.272686832740214
AJ 0.275510877818494
66 0.308907165981276
55 0.339728570797047
44 0.370277078085642
AT 0.376603258153713
33 0.39472290857748
22 0.420782749704168
JK 0.429580617879707
9A 0.467675355645095
JQ 0.488308412300646
8A 0.541147421450351
KT 0.575094314536903
7A 0.611903737296832
6A 0.666299215189457
QT 0.670263371620838
9K 0.686925408736272
JT 0.708201997411991
5A 0.72269869444767
4A 0.768282744282744
8K 0.776496512417781
9Q 0.797516270421039
3A 0.80620992125461
7K 0.839595899667095
2A 0.842448762133094
9J 0.848693921552368
9T 0.86776172504946
8Q 0.876755844542612
6K 0.88712146422629
5K 0.923199448257742
8J 0.925437501040678
7Q 0.926489374056439
8T 0.945891382330858
4K 0.94955376397598
89 0.955820767053449
6Q 0.960353447273573
7J 0.965510376748121
3K 0.967285294166514
2K 0.980013291244393
5Q 0.980534395293884
7T 0.981662856856823
6J 0.985173172709371
79 0.987292261482349
4Q 0.989473335212108
78 0.989810074318745
6T 0.993834463681658
5J 0.994119699008322
3Q 0.994812985764712
69 0.99759687593872
4J 0.997906328821774
2Q 0.9980714879468
5T 0.998194587073906
68 0.998575427785784
67 0.998705028804808
3J 0.99943594677992
59 0.999587690074875
4T 0.999634745650153
58 0.999765084905027
2J 0.999867694241392
3T 0.999917099132857
57 0.999933736436677
56 0.999934409025318
49 0.999983259675907
39 0.999983369642946
2T 0.999983441241245
26 1
27 1
35 1
48 1
29 1
45 1
25 1
28 1
24 1
46 1
23 1
36 1
47 1
38 1
34 1
37 1

Six handed:
AA 0.00441423148229893
AK 0.0247596514320942
KK 0.0277692476420268
QQ 0.054613267027795
JJ 0.0754466066758278
AQ 0.0957609635988554
TT 0.0983964365256125
99 0.124257995924515
KQ 0.140139957548421
88 0.14326888967719
AJ 0.164411984397045
77 0.165971276735913
66 0.185799806320979
55 0.204731776866869
44 0.223738846187826
AT 0.228409752850129
33 0.24339464514137
JK 0.264845841181719
22 0.265926023894547
9A 0.288900311733103
JQ 0.303397375432326
8A 0.343728343728344
KT 0.366640727188436
7A 0.395183315401804
6A 0.44776218220339
QT 0.449064932047857
9K 0.463424337361381
JT 0.476754407788891
5A 0.489800945999212
4A 0.534453476676505
8K 0.542934180272278
9Q 0.56656434082801
3A 0.578161835508185
7K 0.618877857000067
2A 0.619348404255319
9J 0.629041906217806
9T 0.654034331204909
8Q 0.671171245406154
6K 0.681397738951696
8J 0.735921593215305
5K 0.738407684523118
7Q 0.753304769496479
8T 0.780958214687769
4K 0.787651872055542
89 0.79606428228596
6Q 0.811424764759764
3K 0.825665899647392
7J 0.82595762359784
2K 0.860740838866719
5Q 0.861447972772032
7T 0.867776657131992
6J 0.885858953569183
79 0.890896168721695
78 0.902228012158867
4Q 0.904447335935675
6T 0.92576931335789
5J 0.926008706820343
3Q 0.931507305221819
69 0.948943137838642
2Q 0.954437169940194
4J 0.956347690314436
68 0.959671692566842
5T 0.961068438625635
67 0.962138451898024
3J 0.97535689443874
59 0.97809242156393
4T 0.982042574339435
58 0.984816337355788
2J 0.987547813415906
57 0.988268917616563
56 0.990975690975691
49 0.99137444846233
3T 0.992055049531281
48 0.996298438403702
2T 0.996743532523349
39 0.997294443711231
47 0.998226244943969
46 0.998951310861423
45 0.999030781056783
38 0.999191218948585
29 0.999437402786511
37 0.99991716642921
28 0.999917419525327
36 0.999950294916827
26 1
27 1
35 1
25 1
24 1
23 1
34 1

AA 0.000707307369258654
AK 0.00498997032542564
KK 0.00566647483288326
QQ 0.010393369293514
JJ 0.0143203348755232
AQ 0.0193707835147953
TT 0.0198469945355191
99 0.0246499094402969
88 0.0285564350373901
KQ 0.0291890996438333
AJ 0.034042131678003
77 0.0352393617021277
66 0.0412516527104451
55 0.0454808677446657
AT 0.0486322691747251
44 0.0522143489813995
33 0.0533350863353493
JK 0.0573093976783965
22 0.0602463010543103
9A 0.0620425924704952
JQ 0.0690054417042955
8A 0.0792491138145085
KT 0.0842138333581361
7A 0.0935867456092343
6A 0.107308584686775
QT 0.107865761192062
9K 0.114008349347293
JT 0.117181763567185
5A 0.123561216704813
4A 0.137988984841678
8K 0.141036265994829
9Q 0.148619781748332
3A 0.154045651776363
2A 0.16515604249668
7K 0.169326489858565
9J 0.169567010309278
9T 0.182056341297607
8Q 0.188505286365686
6K 0.192626407978961
5K 0.2244346889754
8J 0.224750973019328
7Q 0.231396096933305
8T 0.248140512905088
4K 0.250653475829666
89 0.261829915340219
6Q 0.271896307520011
3K 0.27771124417831
7J 0.280701462489393
2K 0.309192887806964
7T 0.312436603089612
5Q 0.312444851410971
6J 0.332933832709114
79 0.336942126870613
78 0.345710606260732
4Q 0.355731225296443
6T 0.384775305192429
5J 0.387994457457689
3Q 0.393843607778237
69 0.419989728467057
2Q 0.43278976792024
4J 0.439213674647957
68 0.443251102922952
5T 0.44752999586264
67 0.452604287082604
3J 0.491959698405285
59 0.497790525104637
4T 0.514938312770924
58 0.532217114979341
2J 0.545579175131725
57 0.556951075474523
56 0.565466762258866
49 0.579796772265392
3T 0.583123007438895
48 0.621714172277457
2T 0.648449894235854
39 0.658928571428571
47 0.659691593433925
46 0.683275145469659
45 0.694204195248037
38 0.722247903321667
29 0.735777538690722
37 0.772008019485361
36 0.804147657960261
28 0.814530027840382
35 0.828543773572311
34 0.837770507050207
27 0.874080791426216
26 0.926497202344386
25 0.959934383854452
24 0.982678258569965
23 0.991574434140991

[Marc B]

FYI, the reason why you may see some funky numbers (like AA can be dominated), is that a tie counts as domination here, so they are off by some tiny %)