Isn't that clover?

Playing 2/5 NLHE at my local casino.  Effective stack with primary villain in the hand is approximately $1000 (200 BB); the other player in the hand had about $600-700, I believe.  We were playing 5 handed at the time.

I raise to $20 on the button with A:club: T:diamond:.  Only the SB and BB call.  The flop is A:diamond: Q:club: 9:spade:.  It checks to me and I decide to check.  Turn is the 7:heart:.  It checks to me again and this time I bet $40, the SB check/raises to $140 and the BB folds.

I elected to call the turn based on a read.  What opponent types would you call the check/raise against?  What opponent types would you fold to?  The river was the 8:diamond: and the SB bet $200.  Given that I thought that calling the check/raise was going to be profitable, what's my plan for this river card (depending on the read on our opponent, of course)?

If I get some comments about the hand, I'll post my thought process and the results.

I wrote this in a forum post, but I liked it enough that I'll put it here, if for no other reason than as I reminder to myself.

Dear Poker Player,

If you play poker long enough, you will always eventually run worse than you ever thought imaginable. You will experience a run of cards that is worse than anything that has ever happened before and worse than anything that will ever happen again ... until the next even worse run comes. Eventually, you reach a time where you never run bad again. On that day, I hope that you have many friends and family there to remember your life on earth.

So not really sure if I'll keep this up, but the idea of this blog is to talk about the live poker I play in New Orleans.  Games here sound like they different than elsewhere (specifically, the games are generally looser on big streets and more aggressive postflop than what I hear described as 'typical' live play).  I often play both a short stack and a deep stack in the games (maybe I'll elaborate more on this in another post).

Here's a hand from short stack play.  I have about $320 in a 2/5 game.  I'm relatively new to the table, the villain in the hand just made a fold to a turn check/raise the previous hand that he considered to be a 'big' fold (taking his time to muck).  He raises to $25 in the HJ, CO calls, I have AKo in the SB and I elect to flat.  This is one of those weird spots that you get into playing a short stack preflop ... I think shoving will obviously be profitable and one can certainly raise a smaller amount as well, but I think calling is okay here because it allows for a SPR of 3-4.  It's a spot where I am happy to keep weak hands in preflop, because weak Ax and Kx hands can often be induced to stack off with good betting decisions on my part.  Anyway, I think there are good arguments for reraising here, but I also think calling has a fair amount of merit and this spot is a lot closer than most people realize.  As played, the BB calls as well.  The flop comes A83r, it checks to the CO who bets $45, folds to me and I check/raise to $110 total.  CO again tanked for about a minute, then open folded A9s.

Not sure how I feel about my play given that he didn't at least peel a card with A9.  I don't see many players like CO in the games here ... had we played at the table together longer (he left after a few minutes because the game was a must move), obviously he is someone I'd be constantly looking for good spots to bluff him.  And working on not value cutting myself too thin.

Didn't really have any particularly interesting deep hands, but I did 'just' complete AJo and KQo in the SB while playing ~200 BB deep.  As much as I think I understand the importance of position when playing deep stacked, I seem to appreciate it more and more each time I play deep.  I think both hands play better in small pots than in bigger ones, where you can induce a lot more mistakes and you don't put yourself in tough stack commitment situations.  I don't think raising them is terrible in the games I play in, because there is a lot of value in them when some of the looser players are calling any suited cards, dominated broadway hands, unsuited connectors, and weak aces.  When I am playing short, it would be criminal not to make a nice 5-6 BB raise to build a pot, planning to stack off with TPGK; perhaps if I were better playing deep, particularly playing the turn and river, I'd be better off raising them as well.  But I think completing them is optimal for my current playing abilities.  It does make my raising range from the blinds pretty heavily unbalanced toward very strong value hands, but that's fine since I still get action anyway (lol live poker); I could, in theory, add in some raises with some more speculative hands, but since I get a lot of value as is and it's quite expensive (and nowhere near 100% successful) to try to bluff people off marginal hands in my games, I don't see much reason to do so at this point.

Answers to Exercises Exercise 1

(a) The suits of the cards are irrelevant, so the calculation is the same and the answer is 0.274 bets. (b) Whether our opponent has AQ or AK does not affect the calculation since we still have the same number of outs to win, so the EV is again 0.274 bets. © If we miss our straight, we still lose 1 bet, but now if we make a straight, we lose 3 bets. The EV of a turn call is (-3 bets) x (0.182) + (-1 bets) x (0.818) = -1.364 bets.

Exercise 2

(a) The calculation is very similar to Example 3. However, AK, AQ and AJs make up a total of 27 (= 12 + 12 + 3) combinations while AA and 99 make up a total of 4 (= 3 + 1) combinations. Thus the EV of a turn call is [(27) x (0.274) + (4) x (-1.364)] / 31 = 0.063 bets. (b) The calculation is complicated a bit by the fact that we have to break up the cases of when we make a straight based on whether it is the T or the 5 that completes our hand. If we river a T, there are 33 hand combinations that we beat and 6 that we lose to. If we river a 5, there are 39 hand combinations that we beat and 3 that we lose to. The probability of rivering a T is the same as rivering a 5 and it is 0.091 (half of 0.182). We’ll set up our EV equation a little differently this time; you should spend a little thought as to why this method is equivalent to the previous one. Thus the EV is calculated as follows: (probability of not rivering a straight) * (EV of not rivering a straight) + (probability of rivering a T) * (EV of rivering a T) + (probability of rivering a 5) * (EV of rivering a 5) = 0.818 * (-1 bet) + 0.091 * ([33 * (6 bets) + 6 * (-1 bets)]/39) + 0.091 * ([39 * (6 bets) + 3 * (-1 bets)]/42) = 0.131 bets

Hand ranges and combinations

Having worked through these two examples, you may have noticed something particularly unrealistic in them. In general, we will not know our opponent’s hole cards. Rather, we will assign a hand range to our opponent. To do this, we begin by assuming that any two card combination of hole cards is as likely to be dealt to our opponent. We then begin to rule out hands based on his actions. In examples 1 and 2, if our opponent is very tight, we might be able to narrow our opponent’s range down to AA through TT, AK and AQs based on his early position raise. Of course, in practice, we may not be able to definitively pin down a hand range, and we’ll come back to this point later, but let us ignore this issue for the time being. We might assume that our opponent will bet his entire range on the flop, but that when he bets the turn after we call the flop, that he only has AA, AK, or AQs. If that is the case, how do we decide how to evaluate our options? The basic idea is to compute the EV of each option for each of our opponent’s possible holdings and then compute a weighted average of these individual EVs. The reason that we use a weighted average is that some hands are more likely to be dealt than others. A hand combination is the number of ways that a particular hand can be dealt. For example, there are 6 ways to be dealt pocket 4s. This can be computed either by enumerating all the possible suit combinations or by realizing that there are 4 possible choices for the first suit and 3 choices for the second suit, but this double counts the combinations since the order of the suits does not matter, so there are 4 × 3 / 2 = 6 total combinations. Similarly, there are 16 ways to be dealt K7: 4 possible suits for the king times 4 possible suits for the seven. There will be less combinations possible if some of the cards are accounted for. If you hold Ks Js and the flop comes Kh 7d 4d, then there is only possible combination of pocket kings for your opponent, since he must hold exactly Kd Kc to have pocket kings given what you know.

Example 3

We will repeat Example 2 and compute the EV of calling a turn bet, assuming now that our opponent’s hand range is AA, AK or AQs. Let us first determine how many combinations of each hand are possible. Since one ace is accounted for on the flop, there are only 3 combinations of pocket aces possible for our opponent (As Ah, As Ad, Ah Ad). Similarly, there are 12 combinations of AK possible and 3 combinations of AQs possible. Next, observe that our EV calculation in Example 2 applies to each of the AK combinations, not just Ad Ks. Additionally, the EV calculation will be the same for AQs. Thus for 15 combinations, a turn call will have an EV of 0.274 bets. However, we must do a separate calculation for the 3 combinations of AA. You did this in Exercise 1 and should have found that the EV of the turn call in that case was -1.364 bets. The EV of a turn call versus this hand range is therefore equal to [(number of combinations of AK and AQs) x (EV of a turn call versus AK or AQs) + (number of combinations of AA) x (EV of a turn call versus AA) ] / (total number of combinations) = [(15) x (0.274) + (3) x (-1.364)] / 18 = 0.001 bets. This is about as close to a neutral EV decision as one can come up with. Intuitively, this should make some sense. You are very likely to be up against a hand where you have a positive EV turn call (15 out of the 18 combinations), but the 3 combinations where you are drawing dead has a much higher negative EV. In this particular calculation, those two factors balance out almost exactly evenly.

Exercise 2

Repeat Example 3 under the assumptions that our opponent (a) has a preflop range of AA through 88, AK, AQ, AJs, KQs and his range for betting the flop and turn is AA, 99, AK, AQ, AJs, (b) bets his entire original preflop range (AA – TT, AK, AQs) on flop, turn and river and always calls a river raise.

To finish up, let us return briefly to the fact that hand ranges often are not known precisely. We can handle this uncertainty by assigning weighted hand ranges. For example, suppose that we think an opponent could only have AA or AK in a certain situation. We are certain that if he had AA, then he would take the line he has taken. But if he had AK, then we think that there is only a 30% chance that he would play his hand this way. This could an estimate based on our uncertainty (meaning that he either will or will not play AK this way 100% of the time, and the 30% represents a quantification of our certainty that he will play his hand this way) or it could be based on a detailed read that our opponent mixes up his play (so that 30% of the time he has AK he will play it this way, and he will take a different line the other 70% of the time he has AK). While we would still count 6 combinations of AA, we weight the 16 combinations of AK by the 0.30 weighting factor, so we only count 4.8 combinations. (Don’t worry that the combinations are not whole numbers; the math still works the same.) Given these assumptions, the likelihood that our opponent has AA is 6 / (6 + 4.8) = 0.56 and similarly there is a 44% chance he has AK. If there was no weighting, he would have been much more likely to have AK than AA, but since he doesn’t always AK this way, the likelihood that he has aces goes up.

Expected Value

The first tool used to analyze a betting decision is the concept of expected value (or EV). In order to calculate EV, you make a list of all the possible ways the hand can play out; for each possible scenario, you multiply the number of bets you will win or lose in the given situation by the probability of that situation occurring, and then you add up these numbers for all the possible scenarios. A concrete example will hopefully make this clear.

Example 1:

Suppose that a short-stacked early position player raises with Ad Ks; everyone folds to you in the BB and you call with 8h 7h. For the purposes of this calculation, we will assume that the folded SB is equal to the rake paid in the hand, so that the pot has exactly 2 big bets in it at this point. The flop comes Ac 9d 6s. You check, your opponent bets, and you call, so there are now 3 big bets in the pot. The turn is the 9h, you check, and your opponent makes a full bet which puts him all-in. There are now 4 big bets in the pot, and the question is whether you are better off folding or calling at this point. This can be determined by comparing the EV of the two options. The EV of folding is easy to calculate: there is only one scenario where you gain 0 bets and it occurs 100% of the time, so the EV of folding is 0. The EV of calling is only slightly more complicated. There are now two possible scenarios: either you hit one of your 8 straight outs to win the pot or you miss your straight and lose the pot. Let’s calculate the probability of these two events first. There are 44 cards remaining in the deck (52 minus the 8 cards accounted for on the board and in the two players’ hands). Since each card in the deck is as likely as any other to be the river card, the probability of making a straight is 8/44 = 0.182. The probability that you don’t win can be computed in two ways: either as 36/44 = 0.818 (since 44 – 8 = 36 cards cause you to lose the pot) or simply as 1 – 0.18 = 0.82 since the sum of the probabilities of all possible events must add up to exactly 1. If we call and make our straight on the river, we win the 4 bets that are in the pot, while if we call and lose the pot, we lose the 1 bet that we used to make the call of our opponent’s turn bet. (When calculating the EV of the turn decision, we don’t consider how much money in the pot came from our previous bets, as we no longer have control over those bets at this point in the hand. Our decision is the same no matter how the 4 bets ended up in the pot on the turn.) So to determine the EV of calling, we compute (amount won when we make our straight) x (probability of making a straight) + (amount won when we don’t make our straight) x (probability of not making a straight) = (4 bets) x (0.182) + (-1 bets) x (0.818) = -0.09 bets. We interpret this calculation by saying that, on average, calling the turn bet loses 0.09 bets. In general, we write down all our formulas in terms of how much we “win,” subject to the understanding that winning a negative number of bets is equivalent to losing the corresponding number of bets. In this admittedly simplified and somewhat contrived example, the better option is to fold.

Example 2:

Suppose that we keep the action the same as in the first example, but give our short-stacked player two additional big bets. Thus, when he bets the turn, he still has 2 big bets left in his stack. We will further assume that if we call the turn, he will bet any river card and call a raise no matter what card comes. Let us now consider whether folding or calling is better on the turn. (Hopefully, given our assumption that our opponent never folds his hand, you see that raising the turn is clearly worst, though it’s a good exercise to go ahead and compute the exact EV of raising the turn if you assume either that your opponent will reraise you all-in or if you assume that your opponent will call the turn raise and call any river bet.) As before, the EV of folding is still 0. When calculating the EV of calling the turn bet, we must make some assumptions on how we will play the river. Let us suppose that we will fold the river if we do not make a straight (that includes folding if we make a pair of 8s or 7s on the end), and that we will check/raise the river if we make our straight. Given our opponent’s holding and the assumptions about his play, this is the best strategy we could employ. In this case, we will now win 6 bets if we make our straight (the 4 in the pot plus the 2 we win on the river), while we still lose only 1 bet if we don’t make our straight. Mimicking the EV calculation from Example 1, the EV of calling is (6 bets) x (0.182) + (-1 bets) x (0.818) = 0.274 bets. Because of the future money won the river when we make our straight, our turn call is now profitable. This is a prototypical example of implied odds.

Exercise 1:

Repeat Example 2 if our opponent holds (a) As Kd (b) Ad Qs © Ad As. Answers to exercises will be given in Section IV.

Introduction

This is the first in a series of articles that will discuss limit holdem from the ground up. Although the examples will be chosen from limit holdem, much of the discussion applies to all forms of poker. Much of the material covered will be of a theoretical nature. The reason for this is that the alternate approach — giving a concrete set of rules to guide one’s play — can only get one so far in today’s game. There are so many different possible situations that occur when playing poker, and any set of rules will either be too simplistic to give good advice on a consistent basis or too complex for a human mind to remember. Instead, to succeed in limit holdem or any form of poker, there are some key concepts that a player must understand and a thought process that a player must apply in order to beat games, particularly online where the skills of the player pool are fairly sophisticated. In this first article, we will develop the strategic framework for thinking about poker hands. All of the concepts introduced here will be used in the subsequent articles; you should think of the ideas presented here as the basic components of your limit holdem toolbox, which we will use to develop a thorough understanding of how to be successful at limit holdem.

In this article, we will discuss how to compute expected values and how to account for hand ranges and hand combinations when making expected value calculations. When our opponent has a very narrow range of hands, we can actually do the calculations by hand. This article will be somewhat technical, but the ideas presented are needed for a complete understanding of later topics. Future articles will discuss equity, odds, betting theory, position, hand range analysis, hand planning, exploitability, and balance.

The first topic that we will discuss is the luck and skill factors in poker. It is commonly asked whether poker is a game of chance or a game of skill. This is a great example of the logical fallacy known as the false dilemma. The question has no meaningful answer as posed because poker is a game of luck and a game of skill. The luck aspect of poker is immediately evident too us each time we take a bad beat. (Interestingly, our minds tend not to be as sensitive to the luck element when we are the ones who outdraw our opponent.) The skill aspect of poker is a bit more subtle to understand; indeed, many weaker players never acquire much understanding of the skills required to win at poker. The skill in poker arises from the betting decisions made in the course of a hand. The rest of this article will discuss in detail how to determine which betting decision is best in any given situation. Of course, the examples will be kept rather simple and somewhat artificial to make the calculations relatively easy to carry out, but the theory applies to the more complex situations that occur in actual hands.

So I played in a couple of live tournaments this week because the WSOP circuit is here in New Orleans. One thing I really like about live poker is the slow pace of things and the opportunity to really observe every hand closely. I find it helps force me to think through situations more fully and consequently, I often try to learn new games by playing them live first, when possible.

I ended up bubbling in a 6-handed NLHE tournament and then spending five hours folding preflop in a PLO/8 tournament as I ran about as aceless as one can possibly run in that game. Both tournaments were “frustrating” in that luck broke against me (though I also got very lucky in the 6-handed event not to get knocked out early), but I think I had “right view” about them (see Tommy Angelo’s video series if you don’t know what I am talking about) and was able to walk away satisfied that I had played about the best I could. Even though I have certainly tilted and played less than my A game, I feel one of my strengths in poker is carrying a “right view” perspective with me to the poker table the vast majority of the time.

Not really many interesting PLO/8 spots, but here were a couple of hands that came up. Second hand of the event and there’s a min-raise, pot-raise and re-pot raise when I sweat A23Tr in the BB. I folded and the re-potter eventually showed AA23. Two hands later, I overlimp on the button with A228r. Flop comes KJ2r and there’s a bet and raise in front of me, so I quickly fold. Flop gets 3bet and then 4bet. Turn is a K and it goes check / bet / call, putting the players all-in. Of course it’s JJxx vs KKxx. I only had aces once the whole time, and it was AAJ9 with 3 clubs UTG, so I limped, called a small raise, and check/folded on a J83r flop. Not really sure what I should have done there, really crappy spot, but I don’t know a better way to play it. When I was shortstacked (about 10 BBs), I picked up A269r UTG and folded; I’m pretty sure that is probably standard, but at the time the hand looked enticing compared to what I had been seeing. Results oriented, but the pot ended up being a huge pot between AKQ2 and KJJT that was chopped after both parties put in tons of chips with (1) just a low draw and (2) a weak high (just the overpair jacks) and no chance at low. The field was incredibly soft and clueless about the fundamental principle of split pot poker (play to scoop!), but it didn’t matter for me because I never had anything with scoop potential. What I am most happy about it is that during this run of junky hands, I watched every hand play out, tried to get reads on what my opponents were thinking (which was quite challenging, since some of them seemed to think KKQ7ss and AKJ9r were better hands than A642ds), and eventually planning how I wanted to play my shortstack in the most effective manner.