Poker Noir Classroom

*This contains math and reasonably technical poker talk, BE WARNED *

Vegas, Bellagio poker room, 4:45 AM.

You shouldn't play this late into the night -- you start playing like total shit. Everyone gets cocky and stupid after four. You're thinking you'd rather be getting a little cocky and stupid in your hotel suite than grinding out this -- Is that Sammy motherfucking Farha? Wow, fuck that guy. Last time you were here you sat down at the Omaha 150-300 just to see what the big leagues were like. They were like dropping 2500 bucks in 2 hours, but you didn't give a fuck as you did manage to take a big enough hand off Farha for him to give you his fuck-off-small-time smirk and take a break for dinner. Faggot.

Raise it up. You're AH-2H on a 2 hearts flop with no pair. Calls around. You hit the heart on fourth. Raise it up.

The competition is more of the same. Same losers, different table, different night. You're here stag so you're tampon-shaped 3 year old Shuffle is all you've got for camaraderie. One earpiece in, one out, as always. I'll stand in front of you, I'll take the force of the blow. Ugh.

The tables have been softer the past few months. Everyone talks about how the death of online has meant the death of the "casual" player. Casinos are now all sharks, no fish. They say these things. Fucking retards. There's nothing casual about guys who sit at card tables at five in the morning -- some of them are just bad. That's life.

Most of these guys never really learnt to play. They've watched movies and skimmed books. Poker is a math game. Deny it all you want. Play long enough without strong math and your previous good fortune will come back to fuck you. Well, it should and I hope it does.

Poker math starts out simple.

Everyone talks outs in poker. Outs are cards that win you the hand. If you hold two hearts and the board shows two hearts, and you think a flush is a winning hand, you have nine outs from your hearts flush. (Thirteen total - four known cards) This is simple shit. People say "nuts" instead of winning hand. Yeah, that sounds cool. Nuts. "I thought I flopped the nuts." Coolio.

You can figure out your outs with all sorts of hands, but it's meaningless till you can convert the outs back to pot odds. Pot odds describe the ratio of your next bet or call to the size of the pot. If you're on fourth street with nine outs, and you know there's forty-six unknown cards in the deck, you have a nine in forty-six chance of making that winning flush. Nine in forty-six is just under twenty percent, so it's like five to one odds needed to call. That's to say if it's raised thirty bucks to you, you need the pot to hold at least one fifty to make the flush draw call.

If you're landing in deep draws without calling odds, you need to play less fucking spec hands. Spec hands are crap like Ace-X suited or suited connectors. People get seriously wet for that shit. Especially those Rounders wanna-bes. They get off on visions of straight flushes. You get off on visions of -- that can wait.

You're thinking about going into a discussion of implied odds and discounted outs. Fuck it, it's late. That shit is overstated anyways. More excuses to think qualitatively, a.k.a suck at math, and avoid doing any number work.

Most players have two problems. They poorly understand how to calculate pot odds well for when there's more than one card remaining and they fail to incorporate all the social "reading" aspects of the game into evaluating the odds. Well, guess what, it ain't that fucking hard.

Consider this scenario:

You hold 8D 8H.

You raise under the gun. Very tight player early position raises. Loose aggressive calls. You call. Three players see the flop.

Flop: AD 8S 4D

Before going further, we have a made set with a pretty useless backdoor flush draw. If two diamonds fall that don't pair the board, a diamond flush is likely a winning hand, but an 8 is extremely unlikely to be good enough unless we can push out the other diamonds.

Now, before going all math geek faggot at the table, what do your opponents hold? Think back to preflop. Tight player early to reraise means he's holding a top hand, AA-KK-QQ-AK-JJ, maybe AQ or 10s, if he's feeling peckish. Obviously you have to adjust your opinion on what he's likely to play by what you've seen him play, his personality, his recent hands or lack thereof, etc. But honestly, tight, Asian geezers all play the same hands the same way. The other guy, the loose aggressive caller, no matter how reckless, unless retarded, is not calling a geezer without a medium tier hand, minimum, or a spec hand that profits from lots of players and he's hoping for the blinds to limp. Figure the same hands as the geezer, with 6s-9s, A-X suited as minor possibilities. But what about other hands? You have no reason to suspect other hands until you get some better information.

You don't know that much yet. That's why you continuation bet. Your hand is also baller. Books will tell you check raise hands like this -- retarded advice. You learn a ton on third.

You bet, geezer reraises, loose aggressive folds. Call or raise?

The loose aggressive folded. That means he either didn't spike his set (he held a mid pair and didn't hit), he realized his A-X wasn't good (he didn't two pair it), or he played some nonsense garbage that you don't care about. The ace of the A-X you do care about. Now, you put the geezer on a bunch of top tier hands before, let's reevaluate the respective relative chances he holds those cards: (from weakest to strongest)

10-10: Impossible. Could not reraise second to act with the ace on board. Very low probability especially from a tight player.
A-Q: Reasonable. He could very well think this is the strongest ace out there. A reraise will let him know if this is the case.
J-J: Unlikely. Some players like to long play hands like this because they retardedly hate to fold high pairs even in the face of aces. Very low probability.
A-K: Same as A-Q. Very likely hand.
Q-Q: Unlikely. Again, against the ace, a reraise makes no sense.
K-K: Pretty unlikely. Same as with the other high pairs.
A-A: Based on his play, a very reasonable hand for him to have.

Remember the flop: AD 8S 4D

Now, there's once ace on the board. So, there are three aces left in the deck. There are no visible kings or queens. Therefore, there are three ways to have A-A, twelve ways to have A-K, and twelve ways to have A-Q. (simply 3 times 4) Very naively, geezer has, at this moment in time, about 1/9 chance of aces, and 4/9 chances of the other hands. The fact that our loose aggressive came for the ride to the flop but got out of the way makes me inclined to skew those odds slightly more in favor of the A-Q and A-K hands.

If the geezer is rocking the A-A, only the backdoor flush or hitting a fourth 8 takes the hand. Even if we hit the 8, we need him to not hit the fourth A. This is minor, but thinking about it is good practice. Also, you have to decrease the number of unknown cards in the deck when you start making assumptions about other player's hands -- a lot of people don't do this, they're bad and should kill themselves.

Anyways, so if fourth is an 8, any card but an A on fifth wins you the hand, so forty-three cards on fifth would be a win. This shit ain't obvious, so actually think about it. If fourth is a diamond, then any diamond on fifth or that fourth 8 is a win. There are 10 diamonds to go around on fourth, and assuming you catch one, the nine remaining diamonds plus that fourth 8, make ten cards on fifth wins. If fourth is an A, you're fucked and no cards win the hand for you on fifth. If fourth is any other card, only that fourth 8 on fifth is a win. Holy fuck, that was confusing:

If fourth is an 8, 43 wins.

If fourth is an A, 1 win.

If fourth is a diamond, 10 wins. There are ten ways to get a diamond on fourth, so that's a total of 100 wins.

If fourth is anything else, 1 win. (Just the 8) So, 33 total wins. (Think)

There are 45 unknown cards before fourth, and 44 by the time fifth comes around, thus 1980 total outcomes for fourth and fifth street. Of all 1980 outcomes, 178 win the hand for us. That's about 1/11.

Gosh, wasn't that fun and simple? Consider the geezer holding A-K this time instead, but consider losses because you're ahead in the hand. (It simplifies things always to think of the losing hand)

Remember the flop: AD 8S 4D

If fourth is an 8, 0 losses. (Any card wins)

If fourth is an A, 7 losses. (If geezer catches a fourth A, a second K, or the board pairs the 4, 3+3+1=7) Therefore, 2x7, 14 total losses.

If fourth is a 4, 2 losses. (Anything but an A) Therefore, 6 total losses.

If fourth is a K, 4 losses. (Anything but an A or K on fifth) 3x4, or 12 total losses.

If fourth is anything else, 0 losses.

So, 32 losses out of 1980 outcomes, this is about 1/62. The exact same logic can be applied for A-Q and you get the exact same odds.

So, the geezer either holds a hand where our odds of winning are 1/11, 61/62, or 61/62. That's a huge disparity. This is in explicit detail, but practically speaking, rounding off to cleaner fractions is more realistic. Realistically, these odds become, 1/10, sure thing, sure thing. This is the most precise method of calculating your raw probability of winning a given hand assuming you don't get too maverick with rounding off your fractions.

Combining your odds of winning probabilities with how you read the geezer's hand probabilities, 1/9, 4/9 and 4/9 for A-A, A-K, and A-Q respectively, leaves you with 1/90, 4/9, and 4/9 for your odds of winning, or total odds of approximately 8/9 the favorite. Clearly, all the work to calculate the odds out was a waste of time as the odds are in this instance are almost entirely dependent on the hand predictions you previously made.

So, call or raise the geezer's reraise?

You did the math. You know you're currently the favorite until you have information that changes your estimates of what he's holding. With two raises preflop plus the blinds plus the three raises already in the pot, that's 10.5 bets in the pot. Even if he turned over his hand and showed you the aces, you nearly have the odds to call. The issue is: How does my raising here change the probability distribution of the hands the geezer might hold if he calls or reraises my raise?

There are math dorks out there shitting themselves reading this. It's a very strange idea -- logically, it's obvious. You adapt your expectations of the cards your opponent holds based on his actions as the hand plays out -- this is mathematically strange however as you're somewhat arbitrarily changing your probability distribution over time. This is also why, the math above is technically wrong. To be thorough, you need to model how your estimates of what your opponents holds evolve over time -- finding a good heuristic for this is complicated, and this phenomenon is part of the skill of the poker. This is a complicated topic...

Certainly, if you were to raise and the geezer rerasies you back, you have to adjust your estimate on the likelihood he holds the aces. There is no awesome way of modeling this adjustment. A simple heuristic many players use in limit holdem is they simply double the odds of the dangerous hand (in this case the aces) with every aggressive move the player makes. In this circumstance, that doesn't seem necessary, as even with A-K or A-Q, the geezer could justifiably rereaise you again as these are quite strong hands. The biggest shift in the expected probabilities would be an increase in A-K against A-Q as the possibility that the geezer holds the ace with top kicker has certainly improved. In this hand, a called raise does not significantly implicitly strengthen the caller's hand as the difference in kickers is irrelevant to what you hold.

Is raising now as opposed to waiting for the next card pot maximizing? You hold a strong hand and want to maximize its earnings. There's always a ton of discussion on this topic -- most of it is bullshit. Nobody to my knowledge has covered this very thoroughly and discussed the price of information, the correlation between volatility in hand predictions and hand profitability, etc. It's unclear whether a raise here would damage the further earning potential of the hand, nor is it clear that such a raise strengthens the hands the geezer might hold -- therefore the raise is the right play. You gain information and increase the pot size of a pot you're favored to win. Simple, eh?

That's enough for now. There's a ton of sidepoints and whole topics uncovered that are directly related to these ideas, but it can also be a lot to digest if it's not how you normally view the game.

(Back to normal poker noir next time, but I throw this out there just to curiously see how many poker players I've got reading ^^)