Math Players: Does according to bayes theorem...

[Tharpab]

The probability of the SB and BB have a BETTER THAN AVAREGE hand increase after everybody folds to you(in the button), my guess is that it does, or does it?

[TomCollins]

Yes. But not much.

[scottjack]

the probability for the SB, BB having above average hands won't change, according to Bayes' Theorum.

the probabilities that you calculate with Bayes' will differ only by a factor equal to the probabilities that the SS,BB have above or below average hands.

the EP,MP and LP players are folding or calling/raising regardless of the SS,BB's hands.

maybe i'm very wrong on this but to me it seems that it doesn't matter.

[DPCondit]

[ QUOTE ]
the EP,MP and LP players are folding or calling/raising regardless of the SS,BB's hands.


[/ QUOTE ]

Correct, but the fact that none of them found a playable hand would tend to indicate a larger than average percentage of low cards in the folded hands, and a higher than average percentage of high cards in the remaining deck including the small blinds.

What if you had a bunch of rocks who only played AA, KK, QQ, JJ, and AK, and every one of them called around to the blinds? They are not playing their hands based on the blinds, but would this affect the likely hands the blinds could be holding? You bet it would, with almost all of the highest cards missing from the deck, the blinds would almost surely be holding small to medium cards.

Obviously we are not talking anywhere near as pronounced an effect, but a real one nonetheless.

Don

[DPCondit]

I agree with Tom Collins that it probably wouldn't affect it by much, however. Actually the tighter the table, the less the folds mean, and the looser the table, the more the folds would tend to really mean something about the composition of both the stub, and the blinds.

If everybody folded around at a very loose table, it should indicate higher than average cards in the stub and the blinds. However, the blinds would be somewhat less likely to be paired. Since people are more likely to play pairs than unpaired cards, it indicates that the folded hands were less likely to be paired. This means the blinds are also less likely to be paired. This effect is the opposite of the high card/low card effect, the more pairs that are out, the more likely the remaining hands are to be paired.

Don

[tardigrade]

I think this would be true if by dealing one set of crummy hole cards, it made it more likely that the next set would be good. I'm not sure if that's the case, though.

If I get a crappy set of hole cards like A4, then it makes it *less* likely the next guy is going to have AA, and no more likely he'll get QQ than Q2. With suited cards, I would think if I had unsuited, it would be *more* likely the next guy will get unsuited than if I got suited. I'm not sure about connectors.

[playerfl]

I think you are right.

I think the blinds by definition have "average" hands, because there is no hand selection.

If everybody else folds, they had average hands as well.

that doesn't say anything about the value of the blind hands vs. all possible hands, they are still random unselected hands.

[DPCondit]

[ QUOTE ]
I think you are right.

I think the blinds by definition have "average" hands, because there is no hand selection.

If everybody else folds, they had average hands as well.

that doesn't say anything about the value of the blind hands vs. all possible hands, they are still random unselected hands.

[/ QUOTE ]

Except that we know that people call with high cards and pairs (especially high pairs) much more often than low cards or unpaired hands. Because nobody called, that indicates that the folded players were much less likely to have high cards or pairs. This means the blinds are MORE likely to hold high cards, but slightly less likely to have pairs.

Don

[tubbyspencer]

Well if you assume that the folders would have played any Ace(which almost seems true at the tables I play sometimes), it seems there would have to be a much greater chance of the blinds holding AJ-AK, AA.

OK. So no table is quite that bad - but even if you make it A7o or higher - the result is the same. More AJ-AK, AA in the blinds. No?

[imported_stealthcow]

yes, they wlil have better than average hands. but not much considering most players will fold K3 just as quickly as 63.

the reverse should also apply, the more good hands players have in front, the less likely the blinds have strong hands

[DPCondit]

[ QUOTE ]
the reverse should also apply, the more good hands players have in front, the less likely the blinds have strong hands

[/ QUOTE ]

Yes, the reverse is just as true.

[DPCondit]

Sorry, /images/graemlins/smile.gifI just can't seem to stay out of this thread, but let's look what happens to a pair of 5s when eight opponents are holding pairs of jacks or better.

Result
http://twodimes.net/h/?z=433683
pokenum -h 5h 5s - ah ac - as ad - kh kc - ks kd - qh qc - qs qd - jh jc - js jd
Holdem Hi: 278256 enumerated boards
cards win %win lose %lose tie %tie EV
5s 5h 86868 31.22 189460 68.09 1928 0.69 0.313
Ac Ah 4422 1.59 97218 34.94 176616 63.47 0.331
As Ad 4422 1.59 97218 34.94 176616 63.47 0.331
Kc Kh 0 0.00 276328 99.31 1928 0.69 0.001
Ks Kd 0 0.00 276328 99.31 1928 0.69 0.001
Qc Qh 0 0.00 276328 99.31 1928 0.69 0.001
Qs Qd 0 0.00 276328 99.31 1928 0.69 0.001
Jc Jh 60 0.02 270460 97.20 7736 2.78 0.011
Js Jd 60 0.02 270460 97.20 7736 2.78 0.011

Now let's try a lowly 45 offsuit.

Result
http://twodimes.net/h/?z=433700
pokenum -h 5h 4s - ah ac - as ad - kh kc - ks kd - qh qc - qs qd - jh jc - js jd
Holdem Hi: 278256 enumerated boards
cards win %win lose %lose tie %tie EV
4s 5h 87911 31.59 187990 67.56 2355 0.85 0.317
Ac Ah 4892 1.76 98759 35.49 174605 62.75 0.328
As Ad 4920 1.77 98731 35.48 174605 62.75 0.328
Kc Kh 0 0.00 275901 99.15 2355 0.85 0.001
Ks Kd 0 0.00 275901 99.15 2355 0.85 0.001
Qc Qh 0 0.00 275901 99.15 2355 0.85 0.001
Qs Qd 0 0.00 275901 99.15 2355 0.85 0.001
Jc Jh 60 0.02 270033 97.04 8163 2.93 0.012
Js Jd 60 0.02 270033 97.04 8163 2.93 0.012

Because all the big hands have each others outs all copied, a presto (pair of fives) is now worth almost as much as a pair of aces, and so is a 45 offsuit.

Next, using Janne Raveaara's Poker Calculator, let's look at a random hand versus eight opponents each holding S & M group 1 and group 2 hands. This takes a long time to simulate, even on my very fast computer.

Monte carlo simulation results from Poker Calculator 1.1.4.1
Texas Hold'em, 100000 combinations tested.

Hand 1:
Random hand

Hand 2:
Range of hands: AA , KK , QQ , JJ , TT , AKs, AQs, AJs
KQs, AKo

Hand 3:
Range of hands: AA , KK , QQ , JJ , TT , AKs, AQs, AJs
KQs, AKo

Hand 4:
Range of hands: AA , KK , QQ , JJ , TT , AKs, AQs, AJs
KQs, AKo

Hand 5:
Range of hands: AA , KK , QQ , JJ , TT , AKs, AQs, AJs
KQs, AKo

Hand 6:
Range of hands: AA , KK , QQ , JJ , TT , AKs, AQs, AJs
KQs, AKo

Hand 7:
Range of hands: AA , KK , QQ , JJ , TT , AKs, AQs, AJs
KQs, AKo

Hand 8:
Range of hands: AA , KK , QQ , JJ , TT , AKs, AQs, AJs
KQs, AKo

Hand 9:
Range of hands: AA , KK , QQ , JJ , TT , AKs, AQs, AJs
KQs, AKo

Hand | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
------+------+------+------+------+------+------+------+------+------+
Win | 17069| 9144 | 9135 | 9006 | 9166 | 9220 | 9193 | 9103 | 9255 |
Draw | 641 | 3065 | 2999 | 3050 | 2992 | 2903 | 3080 | 3098 | 2968 |
Lose | 82290| 87791| 87866| 87944| 87842| 87877| 87727| 87799| 87777|
------+------+------+------+------+------+------+------+------+------+
Win% |17.19%|10.36%|10.32%|10.22%|10.35%|10.37%|10.42%| 10.35%|10.43%|
------+------+------+------+------+------+------+------+------+------+


Hand no.1:
High Card win: 0 draw: 0 lose: 6690
Pair win: 5 draw: 5 lose: 33566
Two Pair win: 2552 draw: 7 lose: 30266
Three of a Kind win: 2711 draw: 0 lose: 4539
Straight win: 7087 draw: 609 lose: 2858
Flush win: 578 draw: 0 lose: 1746
Full House win: 3637 draw: 13 lose: 2548
Quads win: 413 draw: 3 lose: 72
Straight Flush win: 86 draw: 4 lose: 5


Hand no.2:
High Card win: 0 draw: 0 lose: 9599
Pair win: 843 draw: 560 lose: 36358
Two Pair win: 2663 draw: 1085 lose: 34376
Three of a Kind win: 1476 draw: 28 lose: 2390
Straight win: 76 draw: 1165 lose: 891
Flush win: 1781 draw: 0 lose: 1663
Full House win: 2137 draw: 186 lose: 2475
Quads win: 165 draw: 37 lose: 37
Straight Flush win: 3 draw: 4 lose: 2


Hand no.3:
High Card win: 0 draw: 0 lose: 9734
Pair win: 813 draw: 532 lose: 36566
Two Pair win: 2708 draw: 1115 lose: 34261
Three of a Kind win: 1485 draw: 28 lose: 2265
Straight win: 71 draw: 1143 lose: 926
Flush win: 1719 draw: 0 lose: 1615
Full House win: 2161 draw: 151 lose: 2450
Quads win: 168 draw: 26 lose: 49
Straight Flush win: 10 draw: 4 lose: 0


Hand no.4:
High Card win: 0 draw: 0 lose: 9597
Pair win: 846 draw: 545 lose: 36470
Two Pair win: 2623 draw: 1102 lose: 34541
Three of a Kind win: 1401 draw: 24 lose: 2220
Straight win: 79 draw: 1203 lose: 885
Flush win: 1757 draw: 0 lose: 1645
Full House win: 2122 draw: 150 lose: 2532
Quads win: 170 draw: 22 lose: 53
Straight Flush win: 8 draw: 4 lose: 1


Hand no.5:
High Card win: 0 draw: 0 lose: 9571
Pair win: 868 draw: 509 lose: 36362
Two Pair win: 2699 draw: 1073 lose: 34407
Three of a Kind win: 1456 draw: 25 lose: 2364
Straight win: 74 draw: 1155 lose: 902
Flush win: 1765 draw: 0 lose: 1719
Full House win: 2139 draw: 192 lose: 2475
Quads win: 161 draw: 34 lose: 41
Straight Flush win: 4 draw: 4 lose: 1


Hand no.6:
High Card win: 0 draw: 0 lose: 9595
Pair win: 879 draw: 519 lose: 36345
Two Pair win: 2664 draw: 1027 lose: 34457
Three of a Kind win: 1460 draw: 35 lose: 2341
Straight win: 69 draw: 1127 lose: 896
Flush win: 1813 draw: 0 lose: 1680
Full House win: 2172 draw: 166 lose: 2513
Quads win: 157 draw: 25 lose: 49
Straight Flush win: 6 draw: 4 lose: 1


Hand no.7:
High Card win: 0 draw: 0 lose: 9614
Pair win: 864 draw: 555 lose: 36367
Two Pair win: 2730 draw: 1078 lose: 34359
Three of a Kind win: 1452 draw: 27 lose: 2231
Straight win: 82 draw: 1206 lose: 835
Flush win: 1736 draw: 0 lose: 1705
Full House win: 2174 draw: 189 lose: 2561
Quads win: 150 draw: 21 lose: 54
Straight Flush win: 5 draw: 4 lose: 1


Hand no.8:
High Card win: 0 draw: 0 lose: 9548
Pair win: 801 draw: 546 lose: 36477
Two Pair win: 2635 draw: 1166 lose: 34322
Three of a Kind win: 1472 draw: 37 lose: 2272
Straight win: 65 draw: 1154 lose: 916
Flush win: 1757 draw: 0 lose: 1714
Full House win: 2210 draw: 159 lose: 2506
Quads win: 148 draw: 32 lose: 43
Straight Flush win: 15 draw: 4 lose: 1


Hand no.9:
High Card win: 0 draw: 0 lose: 9626
Pair win: 832 draw: 557 lose: 36456
Two Pair win: 2777 draw: 1037 lose: 34323
Three of a Kind win: 1486 draw: 26 lose: 2309
Straight win: 76 draw: 1146 lose: 873
Flush win: 1765 draw: 0 lose: 1632
Full House win: 2161 draw: 178 lose: 2502
Quads win: 148 draw: 20 lose: 55
Straight Flush win: 10 draw: 4 lose: 1

The point is, given an extreme enough example of enough opponents all holding really strong hands, your random hand, or even two very small cards will hit the flop much more often due to the fact that the stub has become so rich in small and medium cards.

I know I've gotten a bit off track here, so let's get back to the original premise. Let's do three random hands (button and the blinds) against 5 hands lower than group 8 hands (the worst 59.1%) to simulate 5 folders.

Monte carlo simulation results from Poker Calculator 1.1.4.1
Texas Hold'em, 100000 combinations tested.

Hand 1:
Random hand

Hand 2:
Random hand

Hand 3:
Random hand

Hand 4:
Range of hands: Q7s, Q6s, Q5s, Q4s, Q3s, Q2s, J6s, J5s
J4s, J3s, J2s, T6s, T5s, T4s, T3s, T2s
95s, 94s, 93s, 92s, 84s, 83s, 82s, 73s
72s, 63s, 62s, 52s, A8o, A7o, A6o, A5o
A4o, A3o, A2o, K8o, K7o, K6o, K5o, K4o
K3o, K2o, Q8o, Q7o, Q6o, Q5o, Q4o, Q3o
Q2o, J7o, J6o, J5o, J4o, J3o, J2o, T7o
T6o, T5o, T4o, T3o, T2o, 97o, 96o, 95o
94o, 93o, 92o, 86o, 85o, 84o, 83o, 82o
75o, 74o, 73o, 72o, 64o, 63o, 62o, 53o
52o, 43o, 42o, 32o

Hand 5:
Range of hands: Q7s, Q6s, Q5s, Q4s, Q3s, Q2s, J6s, J5s
J4s, J3s, J2s, T6s, T5s, T4s, T3s, T2s
95s, 94s, 93s, 92s, 84s, 83s, 82s, 73s
72s, 63s, 62s, 52s, A8o, A7o, A6o, A5o
A4o, A3o, A2o, K8o, K7o, K6o, K5o, K4o
K3o, K2o, Q8o, Q7o, Q6o, Q5o, Q4o, Q3o
Q2o, J7o, J6o, J5o, J4o, J3o, J2o, T7o
T6o, T5o, T4o, T3o, T2o, 97o, 96o, 95o
94o, 93o, 92o, 86o, 85o, 84o, 83o, 82o
75o, 74o, 73o, 72o, 64o, 63o, 62o, 53o
52o, 43o, 42o, 32o

Hand 6:
Range of hands: Q7s, Q6s, Q5s, Q4s, Q3s, Q2s, J6s, J5s
J4s, J3s, J2s, T6s, T5s, T4s, T3s, T2s
95s, 94s, 93s, 92s, 84s, 83s, 82s, 73s
72s, 63s, 62s, 52s, A8o, A7o, A6o, A5o
A4o, A3o, A2o, K8o, K7o, K6o, K5o, K4o
K3o, K2o, Q8o, Q7o, Q6o, Q5o, Q4o, Q3o
Q2o, J7o, J6o, J5o, J4o, J3o, J2o, T7o
T6o, T5o, T4o, T3o, T2o, 97o, 96o, 95o
94o, 93o, 92o, 86o, 85o, 84o, 83o, 82o
75o, 74o, 73o, 72o, 64o, 63o, 62o, 53o
52o, 43o, 42o, 32o

Hand 7:
Range of hands: Q7s, Q6s, Q5s, Q4s, Q3s, Q2s, J6s, J5s
J4s, J3s, J2s, T6s, T5s, T4s, T3s, T2s
95s, 94s, 93s, 92s, 84s, 83s, 82s, 73s
72s, 63s, 62s, 52s, A8o, A7o, A6o, A5o
A4o, A3o, A2o, K8o, K7o, K6o, K5o, K4o
K3o, K2o, Q8o, Q7o, Q6o, Q5o, Q4o, Q3o
Q2o, J7o, J6o, J5o, J4o, J3o, J2o, T7o
T6o, T5o, T4o, T3o, T2o, 97o, 96o, 95o
94o, 93o, 92o, 86o, 85o, 84o, 83o, 82o
75o, 74o, 73o, 72o, 64o, 63o, 62o, 53o
52o, 43o, 42o, 32o

Hand 8:
Range of hands: Q7s, Q6s, Q5s, Q4s, Q3s, Q2s, J6s, J5s
J4s, J3s, J2s, T6s, T5s, T4s, T3s, T2s
95s, 94s, 93s, 92s, 84s, 83s, 82s, 73s
72s, 63s, 62s, 52s, A8o, A7o, A6o, A5o
A4o, A3o, A2o, K8o, K7o, K6o, K5o, K4o
K3o, K2o, Q8o, Q7o, Q6o, Q5o, Q4o, Q3o
Q2o, J7o, J6o, J5o, J4o, J3o, J2o, T7o
T6o, T5o, T4o, T3o, T2o, 97o, 96o, 95o
94o, 93o, 92o, 86o, 85o, 84o, 83o, 82o
75o, 74o, 73o, 72o, 64o, 63o, 62o, 53o
52o, 43o, 42o, 32o

Hand 9:
Range of hands: Q7s, Q6s, Q5s, Q4s, Q3s, Q2s, J6s, J5s
J4s, J3s, J2s, T6s, T5s, T4s, T3s, T2s
95s, 94s, 93s, 92s, 84s, 83s, 82s, 73s
72s, 63s, 62s, 52s, A8o, A7o, A6o, A5o
A4o, A3o, A2o, K8o, K7o, K6o, K5o, K4o
K3o, K2o, Q8o, Q7o, Q6o, Q5o, Q4o, Q3o
Q2o, J7o, J6o, J5o, J4o, J3o, J2o, T7o
T6o, T5o, T4o, T3o, T2o, 97o, 96o, 95o
94o, 93o, 92o, 86o, 85o, 84o, 83o, 82o
75o, 74o, 73o, 72o, 64o, 63o, 62o, 53o
52o, 43o, 42o, 32o

Hand | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
------+------+------+------+------+------+------+------+------+------+
Win | 12383| 12173| 12336| 8438 | 8652 | 8366 | 8313 | 8321 | 8492 |
Draw | 2994 | 3000 | 3071 | 3258 | 3186 | 3274 | 3296 | 3137 | 3186 |
Lose | 84623| 84827| 84593| 88304| 88162| 88360| 88391| 88542| 88322|
------+------+------+------+------+------+------+------+------+------+
Win% |13.71%|13.49%|13.69%| 9.88%|10.06%| 9.81%| 9.77%| 9.7% | 9.9% |
------+------+------+------+------+------+------+------+------+------+


Hand no.1:
High Card win: 0 draw: 1 lose: 17527
Pair win: 1656 draw: 260 lose: 41887
Two Pair win: 3344 draw: 759 lose: 19247
Three of a Kind win: 2054 draw: 267 lose: 2572
Straight win: 1937 draw: 1325 lose: 1300
Flush win: 1670 draw: 6 lose: 1314
Full House win: 1544 draw: 369 lose: 761
Quads win: 151 draw: 7 lose: 15
Straight Flush win: 27 draw: 0 lose: 0


Hand no.2:
High Card win: 0 draw: 0 lose: 17470
Pair win: 1589 draw: 252 lose: 42146
Two Pair win: 3453 draw: 757 lose: 19269
Three of a Kind win: 1973 draw: 254 lose: 2561
Straight win: 1903 draw: 1341 lose: 1283
Flush win: 1612 draw: 6 lose: 1358
Full House win: 1473 draw: 385 lose: 717
Quads win: 147 draw: 5 lose: 22
Straight Flush win: 23 draw: 0 lose: 1


Hand no.3:
High Card win: 0 draw: 0 lose: 17407
Pair win: 1601 draw: 283 lose: 42016
Two Pair win: 3439 draw: 758 lose: 19302
Three of a Kind win: 1981 draw: 251 lose: 2489
Straight win: 1941 draw: 1373 lose: 1276
Flush win: 1625 draw: 6 lose: 1291
Full House win: 1580 draw: 395 lose: 789
Quads win: 148 draw: 5 lose: 23
Straight Flush win: 21 draw: 0 lose: 0


Hand no.4:
High Card win: 0 draw: 1 lose: 19110
Pair win: 849 draw: 314 lose: 44307
Two Pair win: 2612 draw: 741 lose: 18871
Three of a Kind win: 1505 draw: 304 lose: 2482
Straight win: 1174 draw: 1465 lose: 1315
Flush win: 1069 draw: 6 lose: 1467
Full House win: 1122 draw: 420 lose: 729
Quads win: 91 draw: 7 lose: 22
Straight Flush win: 16 draw: 0 lose: 1


Hand no.5:
High Card win: 0 draw: 0 lose: 19394
Pair win: 874 draw: 291 lose: 43876
Two Pair win: 2720 draw: 767 lose: 18826
Three of a Kind win: 1598 draw: 302 lose: 2587
Straight win: 1096 draw: 1443 lose: 1252
Flush win: 1135 draw: 6 lose: 1475
Full House win: 1118 draw: 370 lose: 730
Quads win: 99 draw: 7 lose: 22
Straight Flush win: 12 draw: 0 lose: 0


Hand no.6:
High Card win: 1 draw: 0 lose: 19207
Pair win: 823 draw: 299 lose: 44134
Two Pair win: 2624 draw: 831 lose: 18893
Three of a Kind win: 1514 draw: 328 lose: 2556
Straight win: 1113 draw: 1447 lose: 1313
Flush win: 1057 draw: 6 lose: 1510
Full House win: 1095 draw: 357 lose: 722
Quads win: 122 draw: 6 lose: 21
Straight Flush win: 17 draw: 0 lose: 4


Hand no.7:
High Card win: 0 draw: 0 lose: 19182
Pair win: 860 draw: 302 lose: 44095
Two Pair win: 2605 draw: 778 lose: 19059
Three of a Kind win: 1459 draw: 315 lose: 2523
Straight win: 1112 draw: 1478 lose: 1339
Flush win: 1116 draw: 6 lose: 1466
Full House win: 1045 draw: 409 lose: 706
Quads win: 107 draw: 8 lose: 21
Straight Flush win: 9 draw: 0 lose: 0


Hand no.8:
High Card win: 0 draw: 0 lose: 19336
Pair win: 877 draw: 308 lose: 44070
Two Pair win: 2581 draw: 759 lose: 18946
Three of a Kind win: 1463 draw: 303 lose: 2566
Straight win: 1179 draw: 1389 lose: 1368
Flush win: 1063 draw: 6 lose: 1509
Full House win: 1051 draw: 366 lose: 724
Quads win: 94 draw: 6 lose: 22
Straight Flush win: 13 draw: 0 lose: 1


Hand no.9:
High Card win: 0 draw: 0 lose: 19380
Pair win: 886 draw: 283 lose: 43806
Two Pair win: 2648 draw: 821 lose: 18983
Three of a Kind win: 1584 draw: 296 lose: 2626
Straight win: 1160 draw: 1375 lose: 1325
Flush win: 1025 draw: 6 lose: 1492
Full House win: 1088 draw: 398 lose: 688
Quads win: 89 draw: 7 lose: 20
Straight Flush win: 12 draw: 0 lose: 2


Don

[Cerril]

I wish I'd paid more attention to proper names in my combinatorics classes. I'd have a better reply to you.

Basically though the answer is 'sort of, yes, but not necessarily'.

While low cards appear the most often in throwaway hands, there are quite a few situations where premium cards could also be thrown away preflop (Paint/xo, AT or KJo for a tight player UTG). At the same time, there are also quite a few ways that a blind could get one of those remaining good cards and still not have a hand (K2, J6). So I'd say that without a great deal of analysis it seems likely that the blinds would be -more- likely to have playable hands, but by no means impossible.

I think this is why you see a lot of situations when everyone folds to the button that the 'position dance' goes raise, fold, call, flop, bet, fold.

[Mason Malmuth]

Hi Tharpab:

This question comes up every now and then and is known as the bunching factor.

For bunching to be significant it would mean that when hands are not played the remaining cards are now more likely to be playable cards. In a game like ace-to-five draw lowball this was certainly the case since high cards are unplayable while low cards are playable.

In hold 'em, discarded hands are often are a mix of a high and low cards. In addition, playable hands include small pairs and small suited connectors. Thus it has always been my opinion that the bunching factor will have very little relevance in hold 'em.

Best wishes,
mason