The Hidden Math of a Limit Hold'em Hand

There seems to be an ongoing debate in the poker world between the "math guys" and the "poker guys." I'm sometimes called upon to speak up for the math guys. Although I'm happy to discuss the mathematics of the game, my arguments don't usually contradict the poker guys. The fact is, to be a successful poker player, you need the poker skills to assess your opponent's hand strength, and you need the mathematical skills to know what to do with that assessment. Poker math is useless unless you know how to apply it. Let's look at a typical hand of limit hold'em, and we'll see how math skills and poker skills come into play.



You're at a ninehanded limit hold'em table, and the first player to act (under the gun, or UTG), a good player, opens for a raise. Everyone folds around to you in the big blind, and you look down at the K Q. It's time for decision No. 1 – call, reraise, or fold? I'm going to narrow the options down to call or fold (the rationale for doing this is the subject for another column). How can we evaluate the two choices? A "math guy" who's unfamiliar with poker might reason that since K-Q suited is almost a 2-1 favorite over a random hand, we have an easy call. But any hold'em player will understand immediately why that doesn't make sense. Our opponent doesn't have a random hand – he has an UTG raising hand. So, what does that mean? Well, when a good player raises from UTG in limit hold'em, it is very reasonable to assume he has a hand in the range of 7-7 to A-A, A-J suited, A-Q, or A-K. Now how does our K-Q suited look? Using PokerStove (software I've mentioned in earlier columns, which is available free at http://www.pokerstove.com/), we learn that K-Q suited has 36 percent equity against an UTG raiser. This is where a player who ignores the math might reason, "I'm a significant underdog, I should fold." Knowing the equity of K-Q suited against an UTG raiser helps only if we can apply it properly. In this situation, we are being offered 3.5-1 pot odds on our call. That means, if the action stopped here, we only would need to win 1/(3.5+1) = 22.2 percent of the time to show a profit. We would have an easy call with our K-Q suited and our 36 percent equity. But the action doesn't stop here. We are not all in. We have to play the rest of the hand, from out of position, no less. Is there enough money in the pot already, and do we have a strong enough post-flop strategy, to make this call with K-Q suited against an UTG raiser and show a profit? Ah, that is the question, and there is no easy answer – for poker guys or math guys. I don't have any kind of proof, but I believe I show a profit by calling in this situation. So, let's say you call, and take a look at the flop.



The flop comes down K 9 3. Decision No. 2 – check or bet? We're going to check (once more, I'll save the rationale for another column). Your opponent bets. Decision No. 3 – raise, call, or fold? To make the right choice here, the first question we need to ask is, do we have any new information about our opponent's hand? The math guy might not know it, but any experienced player will tell you the answer is no. When a player raises from UTG in limit hold'em and is called only by the big blind, the UTG player is so likely to be ahead after the flop that he will bet when checked to, literally every time. So, giving our opponent the same range as before, how does our K Q look after the flop of K 9 3? Well, it looks pretty darn good. Using PokerStove, we learn that our K-Q suited is almost a 2-1 favorite over UTG's range. Armed with that information, the poker guy and the math guy come to the same conclusion: Raise.



You raise and your opponent calls. His call doesn't tell us much. He's getting 7.5-1 on his money, which means he should probably take a card off with hands as weak as A-J suited and A-Q. And if our opponent has a strong hand, such as a set or A-A, he could easily be slow-playing. Maybe we can eliminate 7-7 and 8-8 from our opponent's range, but that's about it.



The turn is the 10. Decision No. 4 – check or bet? Even though it makes a bunch of draws possible, the turn card isn't so bad for our hand. After eliminating 7-7 and 8-8 from our opponent's range, and putting the 10 on board, we're still a 54 percent-46 percent favorite. Bet again.



We do so, and our opponent raises. Yikes. That's bad. Decision No. 5 – call, fold, or reraise? We're not going to reraise, as that would be maniacal. It's unlikely our opponent would make this turn raise with something like J-J or A-Q offsuit. The only hands that really make sense are A-K, K-K, 10-10, 9-9, A J, or A Q. How does our K Q look against this very strong range? Not so hot. We have a mere 19 percent equity. So, we fold, right? Not so fast, poker guy, the math guy still has to speak. We're getting 7.25-1 (two and a quarter big bets went in preflop, two more went in on the flop, and three more have already gone in on the turn) immediately, and even if we assume our opponent will bet the river, too, we're getting 8.25-2 to call him down all the way. So, even if our opponent bets the river every time, and even if we never get an extra bet on the river with a straight or trips, we need to win only 19.5 percent of the time to show a profit, meaning it's about zero EV to call him down. It seems clear, then, that we can't fold here. So, we call.



The river brings the 4. We check and our opponent bets. Decision No. 6 – call, fold, or raise? We're not going to raise. Should we even call? Our hand didn't improve, and if we give our opponent the same range as before, there's only a 9 percent chance that our hand is good. But how often do we need our hand to be good? We're getting 9.25-1 on our money, so we need the best hand only 9.8 percent of the time to show a profit. Based on that, it seems that either calling or folding would be reasonable in this spot. Maybe it's time to ask what the global benefits are to each play – the impact that this decision will have on all the other hands we have to play. If we fold, our opponents may label us as someone prone to making big folds on the river, and we might become more of a target in the future. If we call, not only will our opponents think twice before betting into us, but we'll gain some information about our opponent because we'll get to see his hand. Add into the mix that there might be some chance our opponent is getting out of line with Q-Q or K-J suited, and I think calling is the preferred play.



I arrived at this analysis not as a math guy, or a poker guy, but as a poker-math guy. The numbers in poker are useless without context, and hand-reading ability is useless without the mathematical knowledge of how to use those reads. I'm not a mathematician, I'm just a guy with a bachelor's in the subject. I've made my career as a poker player, and I am much better at calculating hand equities than I am at integrating functions in n-dimensional space. Maybe I would qualify, however, as a "poker mathematician."

Matt Matros is the author of The Making of a Poker Player, available online at http://www.cardplayer.com/.