### MATH Recap, and Bayes Rule in a Poker Context

Nineteen runners came out for the Labor Day edition of Mondays at the Hoy on full tilt, creating a $456 prize pool to be paid to the top three finishers in a field that included two first-time MATHers, one of whom complained about what I think was a pretty fatty turnout for a fucking holiday. Maybe it's just me. Anyways, I got no good starting cards over nearly two hours, receiving pocket Queens and pocket Tens once each, and an AQ and a bunch of JackAces but not even an AK on the night. It was pretty sick. But pretty usual for me, while also usual was me getting to watch pocket Aces and pocket Kings flipped up by my opponents again and again throughout, including a couple of times like on my

**one**pocket Queens of the night which naturally ran into Wes's Aces. Not that I haven't sucked out on Wes and knocked him out of tournaments doing it, don't get me wrong, but when you play for two hours, get exactly

**one**hand better than pocket Tens on the

*entire*night, and that hand runs into pocket Aces, I think anyone has the right to act pissay. Anyways I ended up restealing with 65s in my big blind against Wes's button open-raise with my small stack after we sat with 11 players left for probably a good 30 minutes, which ran into Wes's AK and IGH in 11th place.

Another unspectacular performance for me as my good streak of the past couple of months is now officially

*totally*gone and a thing of the past. The past few days have been among my most frustrating on full tilt for some time. The setups are downright silly, the beats are recockulous. And both plentiful. My one QQ in two hours in the big blind, against the button's AA. That is rich. In the Labor Day 100k on Monday evening, I busted in the

**first hand**when I reraised allin on a K♠Q♠T♥ flop with my J♠5♠, giving me 8 oesd outs and 7 more flush outs for 15 outs twice and therefore making me the favorite in the hand. Still, as I am racing here I do not at all like to call allin in this spot on the first hand of a big mtt, but I love the allin

*raise*since I get good fold equity

*and*am the favorite if called. Unsurpisingly my opponent called with top two pair, unsurprisingly for my luck of late my favored hand failed to make one of its copious outs, and IGH very very early in that one. I ran flopped two pair into a flopflush as well in the 28k to round out what was my third or fourth consecutive super-frustrating night on full tilt. I even started thinking about putting some more money into pokerstars for a bit this weekend, taint-hound that that site is.

Btw did KOD really delete his entire blog? Is that really possible?

Anyways, back to the MATH. In the end it was first-timer Wormmsu ending the night in 3rd place for $91.20 and his first entrance onto the 2007 MATH moneyboard (if you have a blog please leave a comment, you know the drill). In second place was host of the biggest regular weekly blogger tournament out there, Mookie himself, who won $136.80. And winning the MATH this week, fresh back from Tunica for some mtt goodness there was perennial blonkament killer Surflexus.

Here are the updated 2007 MATH moneyboard standings, including the Labor Day MATH tournament this week:

1. Bayne_s $1175

2. Columbo $1168

3. Hoyazo $1162

4. VinNay $775

5. cmitch $774

6. Iggy $745

7. NewinNov $677

8. Pirate Wes $672

9. Lucko21 $665

10. Waffles $650

11. Astin $616

12. Tripjax $561

13. IslandBum1 $527

14. RaisingCayne $522

15. Julius Goat $507

16. bartonf $492

16. mtnrider81 $492

18. PokerBrian322 $490

19. Chad $485

20. scots_chris $474

21. Fuel55 $458

22. Mike_Maloney $456

23. RecessRampage $434

24. Otis $429

25. Surflexus $402

25. Miami Don $402

27. jeciimd $382

27. Jordan $382

29. Blinders $379

30. lightning36 $371

31. ChapelncHill $353

32. Zeem $330

33. OMGitsPokerFool $324

34. oossuuu754 $312

35. leftylu $295

36. Emptyman $288

36. Wigginx $288

38. ScottMc $282

39. Fishy McDonk $277

40. Irongirl $252

40. Manik79 $252

42. Wippy1313 $248

43. swimmom95 $245

44. Byron $234

45. wwonka69 $216

46. Omega_man_99 $210

47. katiemother $209

48. Pushmonkey72 $208

49. Buddydank $197

50. Gary Cox 194

51. 23Skidoo $176

52. Santa Clauss $170

53. Iakaris $162

53. Smokkee $162

55. cemfredmd $156

55. NumbBono $156

57. lester000 $147

58. LJ $146

59. Heffmike $145

60. brdweb $143

61. Mookie $137

61. DDionysus $137

63. Patchmaster $135

64. InstantTragedy $129

65. Ganton516 $114

66. Fluxer $110

67. hoops15mt $95

68. Gracie $94

68. Scurvydog $94

70. wormmsu $91

71. Shag0103 $84

72. crazdgamer $82

73. PhinCity $80

74. maf212 $78

75. Alceste $71

76. dbirider $71

77. Easycure $67

78. Rake Feeder $53

So that's 78 different bloggers and non-bloggers alike now who have cashed at least once in Mondays at the Hoy during 2007, through by my calculation approximately 32 MATH tournaments, give or take probably 1 tourney. This week Mookie and wormmsu enter the moneyboard for the first time, while Surf jumps into the top 25 with his win in addition to previous cashes this year in the Hoy. No movement in the top few spots or even in the top 25 at all this week, so that virtual tie for the top spot between myself, Bayne and Columbo will persist for at least one more week. Congratulations again to all three cashers and I look forward to another fun time next Monday in Mondays at the Hoy on full tilt.

I kinda want to get this post up for today, but I know I talked a lot about Bayes Theorem last week and a number of you have been asking about the promised discussion about how this relates to poker. I'm thinking I will write some more about this later in the week, but I will start that discussion here today with a quick example poker problem for you all to chew on. I'm going to try to give a hypothetical that applies somewhat to my own style of play to illustrate a point from

*The Mathematics of Poker*that I've been reading for the past week or two.

So we're talking about a typical aggressive player here. What percentage of the time is such an aggressive player dealt what most of us would consider a "strong" preflop raising hand? Let's assign a 0.45% chance to being dealt each of AA, KK, QQ, JJ and TT, and 1.2% chance of being dealt AK. That totals to 3.95%, so I'm going to go ahead and use 4% just to make that calculation easier. So let's say that this aggressive player is dealt a "strong" raising hand 4% of the time in nlh, and he will put in a preflop raise 90% of the time when he does receive such a strong raising hand preflop. Now, of this aggressive player's 96% of hands that are not the strong raising hands, assume he will also raise with those hands about 20% of the time (probably somewhat typical of an aggressive preflop player).

The question is, then, if this aggressive player puts in a preflop raise, what are the chances that he actually has one of those six "strong" raising hands? I suggest that you re-read the assumptions above, and make an estimate based on the percentages you see as to what you think the correct answer is, and then read on below where I will use Bayes rule to come up with the mathematically correct answer.

OK if everyone has come up with their estimates or answers to the question, here is how to solve it by using Bayesian mathematics. The chances that a preflop raise from the aggressive player we described above actually indicates one of the six strong raising hands TT-AA and AK are:

__90% x 4%__

(90% x 4%) + (20% x 96%)

This equals:

__.036__

.228

Which equals

**15.79%**. So there you have it, even though an aggressive player raises 90% of the time with his strongest raising hands, those strong hands occur so infrequently that a preflop raise from such an aggressive player only generally means a 15.79% chance, or less than one-sixth of the time, that he actually has TT-AA or AK. In other words, if you understand the theory of conditional probability sufficiently, then there is a more than 5 in 6 chance that when such a player raises preflop, he still does not have a "strong" hand.

That's the lesson for today on Bayes theorem in a poker context. I plan to write more about this later in the week, but it is my contention that without a doubt most players do not appreciate the significance of Bayes' rules on conditional probability, and that most players I play with with some regularity do not properly make this calculation, even if it is only being estimated subconsciously. In other words, most people do not respond to the aggressive preflop players by treating their preflop raises as a more than 5 in 6 chance of not really being among the top hands. Most players fold far too often for that. It's just something to think about -- later in the week I will write some more about the implications of Bayes rule in this context, as well as some other areas where it can be used to make useful conclusions based on conditional probabilities in the world of poker.

Labels: Bayes Theorem, KOD, MATH Recap

## 11 Comments:

Excellent. But how do you adjust this example for Astin?

I am amazed that you are doing math problems here. The fact that your weekly tourney is called "MATH" is the funniest thing around.

Now for your problem, I don't think that you need to do that much math to know that someone who is aggressive preflop is raising a bunch of hands that are not in the top 4%. A simple approximation is that someone who raises their best 24% of hands (similar to your example), only has a top 4% hand 1/6 times. No need to dig up Bayes for this one. Also Should'nt the aggro player raise ALL of his top 4% hands. What does raising 90% do for him if he is already raising 24% of hands?

want to hear my ftp suckout stories examples from last three days.

Sunday

AA vs KK (I reraise he reraises me and I of course call knowing he has kk).. he has a k of diamond, one diamond on flop turn and river both diamonds.

Saturday

I flop broadway 2nd to act. dude bets pot with top pair K. I repop 3x.. he calls. 9 on turn he two pairs and moves in.. I call he rivers a 9.

Friday

My flopped two pair getting rivered by a better two pair when the board pairs.

Over and over and over and over all month and now into september.

I got AA once in the last week and im screwed by it.

Its all good though.

What is the chance that a Blinders raise is a top six hand?

I'm no math wiz like most of you Einsteins but I'm saying 95%, slightly lower than Smokkees 100% and way less than KODs 2%.

Goat - it's easy, I only have premium hands, therefore the answer is 100%, or 6 out of 6.

I think the math is too simple. Assuming you're playing a 9-man (or even 6-man) table, you're competing against more than one player, and if you have multiple aggressive players, than they each have a 15% chance of having a premium hand. Even heads-up, you're still gambling that their non-premium hand is worse than your non-premium hand.

Which is why poker is more than just math. Besides, an example like this should be intuitive by this point for most of the blogger community.

Unless, like I said, I'm the aggressor, then you should just fold.

On my way to work but wanted to add a thought...

'Randomizing' play. To make EP preflop raises and limps correct, one must randomize their play so as not to allow their opponent to 'read' or 'put you on a hand' so easily. Which is why one may limp AA UTG every once in awhile.

Does the Bayes also take into account a play such as raising UTG with a hand such as 78s?

That's great if all the players at your table tell you that they're aggressive and will raise x amount of the time!

What Bayes is actually useful for is to help you weight evidence.

wormmsu said...

hoy,

i am not a blogger just play poker everyday when im not working. I read your blog everyday and really enjoy all that you write. And your tournament always sounds like a good time so i thought i would give it a go. it was a blast and filled with fun donks, so i will be back when im not working sincing i run a bar.

worm

5:04 AM

i had a great time playing the MATH and will try to make it back weekly ,

Thanks for the post, given me something to occupy my mind during the dull spots in my day...

ok, two comments (possible corrections)

1. Bayes Theorem is not to be confused with BAYNE'S theorem, which is much less than 15% ;)

2. I think when you do your calculations, you are assuming a "closed set". You can't do that if the player has assigned some hands as unplayable, or even some situations as unplayable.(?) I think you need to subtract the finite universe of times the player will not play a hand, say closer to 70%. This decreases the variance.

Moving forward from there, can we then state that the fold value is added to adjust the P(B/A).(?) Lets say its (34% of 90%) about 30%.

Now that has to be adjusted for. When a player RAISES pre-flop then, assuming he plays about 33% of his hands, he has a stronger hand closer to 30% of the time?

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