You Just Called With That?

Recently, I was involved in a very interesting seven-card stud hand. The end result was that I had kings full and was check-raised on the river by a board that was A-4-J-8 rainbow, and I merely called. As you might guess, it turned out that I had the best hand. What? I only called with kings full when my opponent had only one ace showing, no open pair, and zero straight-flush possibilities? It sounds crazy, doesn't it? As I raked in a sizeable pot for a totally heads-up hand, I glanced up to see other players looking at me as if I were just that - crazy. Perhaps they wondered if I was stupid or somehow misread my hand.

Third street: Let us start at the beginning and see how I ended up at this weird river scenario. I am playing fourhanded $40-$80 stud. The low card opens for $10 and the ace makes it $40. I am rolled up with trip kings, and simply call. At this juncture, I am positive that I have the best hand, as it is almost impossible that my opponent (who has played stud longer than I have walked the earth) would have made the classic error of a neophyte, raising with rolled-up aces, particularly in a shorthanded game and from early position. I decidedly put him on a pair of aces. The low card folds.

Fourth street: It brings me an offsuit 6, and an offsuit 4 lands on top of that ace. He bets, and again, I just smooth-call, all the while mentally planning to raise on fifth or sixth street. I simply do not wish to give away the strength of my hand at this point, nor do I want to lose his action should his raise have been nothing more than a pure bluff.

Fifth street: He catches a jack that does not suit up with either the ace or the 4. I catch a 10, which also does not match suit with either of my boardcards. The momentary dilemma I face is whether to raise on fifth or sixth street. The risk of raising prematurely is winning a relatively small pot right then and there, especially considering that in a shorthanded game, there is a good chance the ace raised with nothing. However, there are two risks involved in waiting until sixth street to raise. The first one is that he simply does not bet out on sixth street after my fifth-street call. The second one is the one-in-six chance of pairing up openly (seven of 42 unseen cards will pair me), losing my position, and probably not getting the raise in unless I attempt a check-raise, which, arguably will work only if he's made aces up. Within this framework, my hand does not really look like a straight or flush draw (yes, you could argue that I could have started the hand with J-Q-K, caught the blank 6 on fourth street, and come back on the straight draw on fifth, but I would be somewhat foolish to play that hand with his ace being a card that completes my hand, and even more foolish to have the audacity to raise with it). So, if that open pairing occurs, I must assume that my opponent has the cognitive reasoning to perceive it as two pair, and I will have almost no shot at a check-raise if he has only aces, and I will hate missing a bet altogether with kings full. At any rate, he leads out with $80, as expected. I decide from his seemingly genuine forcefulness that he has at least those aces for sure, and that I can safely raise him without losing his action. I do so, which should tell him that I have his aces beaten. Much to my delight, he three-bets it, which silently tells me that he can beat my possible two pair (which are probably thought to be kings up at this precise moment in time). I make another raise, for a total of $320, which in turn informs him that I can even beat his aces up, and he must therefore revise his assessment of my hand and put me on trips. Unless he has trip jacks, he can only call. Such is the back-and-forth dance of silent telegraphed messages that often accompanies a heads-up pot.

This particular street deserves further analysis. Actually, there are several lessons that can be learned from it. His call of my four-bet (as opposed to a reraise) absolutely indicates aces up or trip fours, either of which are virtually interchangeable from the standpoint of how he would probably play them. On the surface, you might surmise that in my spot, you would paradoxically hope that he had one of the higher hands of trip fours or jacks, since that almost entirely removes the possibility of aces full on the river. As it turns out, though, mathematically you would still want him to have aces up, which is roughly a 5.5-to-1 underdog, and not the trips, which is an almost exact 3-to-1 underdog. These odds are explained in later paragraphs.

Rolled-up aces were more than 99 percent out of the question, for obvious reasons, and I figured that trip jacks were not a very real probability, either. I did so by momentarily putting myself in my opponent's seat and asking myself what I would do. Let us all put ourselves in that seat for the moment. I submit that correct play would be making a fifth bet, for a total of $400. The reason for this would be the following: All of the betting, calling, and raising on the part of the player showing K-6-10 could and probably would play identically if he had started the hand with pocket sixes, pocket tens, or rolled-up kings. So, there is a one-third chance of trip kings, but a two-thirds chance of trip sixes or trip tens. Making it five bets with trip jacks would be correct for two reasons. Initially, it is due to the simple laws of probability. Of course, one could argue that probability would dictate that two-thirds of the time you would gain that extra (fifth) bet, but that one-third of the time you would lose two extra bets (your fifth and the opponent's sixth), which would make for a break-even proposition. In truly break-even propositions, I opt for taking the aggressive role, since at that moment in time, you are the 2-to-1 favorite not to face the sixth bet. Now, if the opponent does put that rare sixth bet in, you can be very sure that you are facing trip kings. The true value of making the fifth bet is gaining information. If the opponent just calls, you can be very sure that you are currently in the lead, and therefore can safely bet out on sixth street if he does not make an open pair. If the opponent raises you, you know what you are up against. Either way, the fifth bet should provide you information, but probably should be made only with trip jacks.

While we are still collectively in the opponent's seat, let us consider one other option. If you have aces and jacks and face an opponent who has just made it four bets with K-6-10 offsuit, you can be quite sure that you are facing trips. (Note: It is remotely possible that an opponent could make it four bets with only kings up, thinking that you are capable of a semibluff three-bet with just the aces, but this is simply a highly unlikely scenario in a small- or medium-limit game, and only slightly more likely in big-limit poker.) Let us get back to more realistic and common scenarios. Folding at this juncture is not a bad idea at all. Granted, folding aces up is an emotionally difficult thing to do, but let us view this scenario logically, clinically, and unemotionally. Assuming that the opponent's fourth bet is a sure-fire indicator of trips, there seems to be an equal chance of any of the three possibilities (sixes, tens, or kings).

Purists would state that the aces-up hand has a statistical possibility of filling up 18.36 percent of the time. They would arrive at this by calculating the odds of missing the hand on sixth and seventh streets, and subtracting that from 1 (or 100 percent, if you prefer) to yield the odds of making the hand. They would assume 42 unseen cards by subtracting their own five cards, the opponents visible three cards, and the folded doorcard of each of the two other players in the fourhanded game. The calculation would be: 1- [(38/42)(37/41)], or 18.36 percent. I am going to add a little intuitiveness into the equation for a more likely percentage. Since we can safely assume the opponent has either trip sixes, tens, or kings, we can safely say that we have "seen" two more cards that will not help our hand (either two more sixes, tens, or kings). With two "streets" to come and 40 unseen/quasi-unseen cards left in the deck (52 minus your five cards, your opponent's "five" cards, and the two folded doorcards), we calculate: 1- [(36/40)(35/39)], or 19.23 percent. Well, an almost one-in-five chance of filling doesn't sound too bad, does it? Let us look further. Consider that only half of that will it be aces full, or 9.615 percent. Let us now look at the opponent's hand to see how grim this really is.

Utilizing the same intuitive logic about unseen cards, the trips (whichever they may be) have a statistical likelihood of filling or making quads calculated as: 1- [(33/40)(29/39)], which is a whopping 38.65 percent. That's almost 3-to-2 odds! That is also just over twice as likely to fill than the two pair. Of that, the likelihood of quads accounts for exactly 5 percent. Also, let us remember the one-third possibility of any full house being that of kings full, or 12.88 percent. That is more than the aforementioned 9.615 percent chance of making aces full. We must also consider that the opponent will make quads 5 percent of the time. Aces up does not appear to be as healthy a hand as one may have initially considered.

According to my computer analysis of running 2.5 million hands of the aces and jacks vs. the low trips, you have only an 18.43 percent chance of beating either of them. An even smaller chance of 15.46 percent exists to beat the high set, since filling up on one side will lose to him should he fill as well. This averages out to 17.44 percent, [(2x18.43)+15.46]÷3, making you a 4.75-to-1 underdog. So, on fifth street, facing that fourth bet, there is $770 in the pot (four $10 antes, one $10 bring-in, two $40 bets on third, two $40 bets on fourth, and your $240 and his $320 on fifth). Sure, you could argue that you're getting a 4.75-to-1 shot at an almost 10-to-1 pot (770-to-80), so you have to call, right? But are you truly getting those pot odds? I would think not, for filling the small side is hardly a cinch hand. What about the call you must make on sixth street as well? How about the river? Your opponent will likely bet unimproved trips as well as the full house. Would you be prepared to lay down the aces up then? If not, you are not calling just one more bet on fifth street; you are calling three more bets: fifth, sixth, and seventh streets. Folding right then and there would not be a bad play whatsoever. In fact, I would say that it is the correct play. Nevertheless, the call was made.

Let us now return our collective perspective to viewing the hand from my seat, which holds the same three kings I started with. What do we know so far? Our opponent's hand simply cannot be trip aces, and it is almost as unlikely that it is trip jacks. Trip fours is a very real possibility, but aces up is by far the most likely hand, if for no other reason than the simple probability that with five cards, you are going to have two pair about 2.17 times for every one time you have trips. Plus, there is that intangible feeling, instinct, hunch, or whatever you want to call it that just "told me" it was aces up. The instinct of a player is a factor that should not be discounted, and as players, we should strive to better ours on an ongoing basis.

Sixth street: It arrives rather uneventfully. I catch a 9 and my opponent catches an 8. Naturally, he checks after all that fifth-street activity. I bet, and he calls. This mere call solidifies the certainty in my mind that he has aces up, and I am pretty sure it is aces and jacks (I thought I "saw something" when the jack hit him on fifth street), although the rank of his second pair is not altogether relevant, since my trips are kings. If this concept of irrelevancy is not immediately clear, it will be made so subsequently. Furthermore, I reiterate that there is extremely little doubt in my mind about the fact that he has put me on trips of some sort. Why else would he have stopped raising on fifth street and only check-call now? Possibly, he is giving me trip sixes or tens, but he has to consider rolled-up kings, as all three of these possibilities would probably play identically in this context.

Seventh street: I manage to catch another 9 to fill. My opponent predictably checks, and, of course, I bet $80. To my surprise, he makes it $160. My knee-jerk reaction is to raise again. After all, I have kings full, for Pete's sake! But therein lies the question: Is it correct to raise?

My analysis is as follows: Due to the four bets placed on fifth street, with mine being the final one, coupled with the mere check-call on sixth street, I'm virtually 100 percent positive he's going to the river with aces up. That being said, his check-raise in this spot metaphorically spray paints the phrase "I filled up on the river" in florescent pink letters on the tropical mural wall in the Mirage poker room. But which full house does he have? Since I have seen no other aces out, nor have I seen any of his other cards out, especially jacks, there is exactly a 50 percent chance that he has aces full and a 50 percent chance that he has a smaller full house (again, probably jacks, but it's not relevant, since all full houses except aces full are beneath my hand). So, if I elect to put in three bets, I absolutely tell him that I have improved my trips to a full house of my own. If he's got the small full house, chances are that he just calls and I win one extra bet. However, if he's got the aces full, he will reraise me for sure, and that will cost me two extra bets. Granted, he could put in four bets anyway with jacks full, since I very well may have only sixes or tens full, but that would still leave me in a spot in which I couldn't raise for the following reason: If I raise from either the second bet or the fourth bet as my "starting point," I make only one more bet 50 percent of the time, but lose two more bets 50 percent of the time. Also, would he four-bet with jacks full? I submit that since he has played the game for many years, he would probably reason that I would not three-bet with tens or sixes full because the threat of his having jacks full is certainly there. In fact, his making his second pair on sixth street is extremely unlikely due to all of the action on fifth street. So, he had either aces and fours or aces and jacks. His full house (from my perspective) has a two-thirds possibility of being jacks full or better and only a one-third possibility of being fours full, so I most likely would not three-bet it with anything except kings full; ergo, he most likely would not four-bet it with anything except aces full. In my opinion, the play is a 2-to-1 dog to raise, so the correct play is to just call.

This hand illustrates, among many important concepts, the absolute necessity and complexity of not only considering what your opponents' most likely hands are, but of momentarily "getting inside their heads" and attempting to ascertain what they are thinking your most likely hand is. Deeper levels of thinking what they're thinking you're thinking what they're thinking, and so on are required in certain hands, and this certainly was one of them.

Many aggressive professionals would argue that poker is all about earning an hourly rate commensurate with the size of the game, and therefore in situations like this one, you should go for the extra bet. What this fails to address is that hourly rates are made up of two factors - bets earned and bets saved. Out of 100 theoretical trials, I would make 50 extra bets when he had the small full house and lose 100 bets when he had the aces full. This is a net loss of 50 bets per 100 trials, or, simply put, a negative expectation of one-half a bet per event. It should be clearly obvious that any negative expectation will diminish your hourly rate. It's that simple. There is an old poker adage that states, "Whenever one person bets and the other calls, somebody made a mistake of some sort." Hopefully, this proves that saying to be patently false.