Hand Analysis

The following scenario is a familiar one for almost any limit hold'em player, especially one who likes to play shorthanded. Play is folded around to the button, who is known as a tricky player and an aggressive blinds stealer. He open-raises and you are in the small blind with a very decent hand – the A Q. You know all about play from the blinds and make the obvious and correct play of three-betting in order to put maximum pressure on the big blind. You are hoping to get heads up with the button, as you almost certainly have the better hand.



The plan succeeds admirably. The big blind does indeed fold and the button declines to confuse the issue with any further raising. There are seven small bets in the pot and you are very happy with the progress of the hand thus far. The arrival of the flop slightly dampens your enthusiasm, because – as usual – it misses you completely. It consists of three random cards lower than a 10, and for the moment we will not worry too much about what these cards actually are. "Never mind," you think, "I still have my two big cards and the flop probably missed him, too. I must keep up the pressure." Indeed you must, so you bet, and the button dampens your enthusiasm for the hand still further by raising. There are now 10 small bets in the pot.



With pot odds of 10-1, you have an easy call. Even if you are certain that your opponent has a pair, your overcards give you six outs (probably), which makes you about 7-1 against improving on the next card. Even if you downgrade your outs (your opponent may have an A-X or K-X hand in which X makes a pair; there is also the possibility of redraws on the river), you certainly should count them as being worth around four and a half or five outs, and you still have value for a call.



However, you know full well that your opponent is tricky and could easily not have any pair at all; you may even have the best hand right now. Considering this, you call, and there are now 11 small bets (five and a half big bets) in the pot.



The turn is another random low card. There now is no reason to bet, so you check, and the button bets, increasing the pot to six and a half big bets. Now you have to decide what to do, and there are various factors to consider:



1. If you are 100 percent sure that you are just playing overcards, you do not have pot odds to call and should fold.



2. If there is a reasonable chance that your opponent – who is known to be tricky – is pushing a draw or even completely bluffing, you may have value for a call.



3. If you call on the turn and the river is a blank, you will have to call again if the button bets the river. It would not be logical to fold, unless the board becomes very gross (however, if there is a reasonable possibility of this happening, you probably should be inclined to fold on the turn in the first place). By deciding to call on the turn, you are making a clear assumption that there is a small possibility that your opponent is bluffing/semibluffing, as you do not have pot odds to call purely on the value of your overcard draw.



So, now your job is to weigh all of this evidence and decide what to do. Some players take a rather simplistic view of this situation, deciding one of the following:



1. I may well be winning the hand and I have outs even if I'm not. I am not going to be pushed off what is possibly the best hand. I call.



2. It will cost me two big bets to find out if he is bluffing. That is just too expensive, and does not represent good value. I fold.



If you always call here, you will be making a mistake, and if you always fold, you also will be making a mistake. If the game is very shorthanded (four players or fewer), you certainly need to call some of the time to prevent other players from taking a shot at you every time a similar situation arises. So, what should you do?



Dan Harrington has an excellent section in his no-limit hold'em tournament book (Volume 2) that deals with a topic he calls "Structured Hand Analysis." This is a method of estimating the probabilities of various outcomes in a hand, and therefore deciding upon what is hoped to be a profitable line of play. Of course, the technique he decribes deals specifically with no-limit tournament situations (usually when someone has made a bet that will put you all in if you call), but the method can be adapted to limit play.



Before we deal with the calculations, we must be aware that in a live tournament situation in which all of your chips are on the line, no one is going to object if you think for a minute or two before reaching a decision. However, in a typical online game, you will have only 15-20 seconds to decide what to do, so we must try to keep the calculations simple, while ensuring that they do not become oversimplified and thus rendered useless.



The following line of reasoning is quick and dirty. It is a long way from being mathematically exact, but I have tried to make it sufficiently simple that – with a bit of practice – it can be done in the 15 or 20 seconds that you typically get in an online game to make your decision. OK, here goes. First, we are going to work out how often we need to win to get value for our bets (well, calls, to be precise), and second, how often we will win when we play out the hand.



How often do we need to win?

The pot contains six and a half big bets. If we go on to win the pot, sometimes we will win one more bet and sometimes not. So, let's say the average result is a gain of about seven big bets. It will cost us two big bets to see the hand through – one to call now and one to call on the river. So, we are risking two big bets to try to win seven. Thus, if we win two times out of nine, we will be at the break-even point. Let's turn this into a percentage. Two out of nine is going to be a bit more than 20 percent, but let's keep it at 20 percent, because sometimes the river card will be scary (although unhelpful to us) and our opponent will check the winning hand. So, we need to win 20 percent of the time to break even.



How often will we win?



We have two ways to win:

1. We are behind but we win with our overcard draw. Now we need to decide what our overcards are worth in terms of outs. Clearly, there is a maximum of six and possibly none at all if our opponent happens to have stumbled into a big hand. Much of the time, our overcards will be worth six outs, so let's say four as a reasonable compromise. Four outs with one card to come is in the region of 10-1, and is therefore worth about 10 percent.



2. We are ahead and our opponent does not draw out on us. What is the chance that we are currently in the lead? With a tricky opponent, this should be at least 20 percent, and could be much more. However, he will have outs; a reasonable estimate would be to give him 10 outs – the equivalent of a gutshot draw (remember, he also can win by simply pairing). Ten outs give him about a 25 percent chance to improve and overtake us (I know that this isn't exact, but we haven't got time to fiddle around with pocket calculators). Thus, we need to downgrade our 20 percent chance of winning to around 15 percent.



Now, we are nearly done. We add the 10 percent and the 15 percent and get 25 percent. We needed 20 percent for a calling-down strategy to be correct, and we have exceeded that. Thus, we call. This may seem surprising. After all, we know full well that a mere couple of overcards should not be worth a great deal on the turn. However, the pot – at six and a half big bets – is already quite large. With less earlier action and a pot size of four and a half big bets, the previous line of reasoning remains valid, but now we need to win two times in seven to break even. This works out to be about 29 percent, and we should be more inclined to fold.



This may seem frighteningly complex, but it isn't, and it gets easier with practice. It also is a more satisfying way of determining your play rather than simply trying to decide if your opponent is bluffing or not.

Byron Jacobs is the author of How Good is Your Limit Hold Em? with Jim Brier. It is available through bookstores and at http://www.dandbpoker.com/. Byron may be contacted at [email protected].