I fully realize that using the word "math" in the title of an article isn't necessarily the best way to get a lot of people to read it, but for the serious Hold 'em player, such articles are downright crucial if you want to optimize your profits from the game. Uh, oh..."mathematics" and "optimize" together in the same sentence. Okay, I promise it's really very simple, as was my premise for writing this lesson. I've already covered the basic math of the Small Blind (SB) bet in Lesson 6 by showing you which hands are worth playing from a profits point of view, but that listing is based upon averages and, while averages are fine for the "average" player (pun absolutely intended), neither you nor I are average players. What sent me off on this tangent was seeing so many players just calling the Big Blind (BB) bet, so by the time the bet came around to me in the SB, I might have to bet only another $5 (in a $10/$20 game) to see the flop when the pot was $45 or more. (With me on this? The Big Blind put up $10, I put up $5 in the Small Blind and if three other players "limp" by betting $10 each, the pot would be $45 when the bet got to me, assuming the BB limps also.) You need to remember that my $5 SB bet is no longer mine; if I fold it's gone. But if I do complete the bet, I'm getting 9 to 1 "pot odds", which is very healthy, but again, I'm assuming the BB does not raise. If the BB does raise, the pot odds may not be so favorable and folding may be the only proper play.
Is a 9 to 1 return on investment healthy enough to play some of the lousy hands we all seem to get when we're in the blinds? That was the question I wanted to answer. In my Limit Hold 'em Basic Strategy Matrix (see Lesson 11), I list the hands that should or should not be played from the SB, depending upon whether or not there's been a raise, etc. But, as I mentioned earlier, that's all based upon averages - more of a situation where 3 players or less limp into the pot, or someone raises ahead of you - rather than a situation where 4 or 5 or even more players just limp. Of course, just because a player limps into a pot, that doesn't necessarily mean they're holding a lousy hand. In fact, it's been my experience that many players will limp with A-K or even A-A in limit games because they don't want to scare out the other players before the flop. (Those are also the same people who moan, "I never win with Aces", after someone with 7-4o draws out on them, something we want to do if the math makes sense.)
So, with all of this in mind, I began analyzing various scenarios. Needless to say, I couldn't cover every possible combination of hands one could be against with 4 or more players, but I could make some "worst-case" calculations that seem to make sense. For example, 7-2 offsuit is frequently called "the worst hand in poker", which isn't completely accurate. It may be the worst hand pre-flop, but if the flop comes 7,2,2, you're in pretty good shape, don't you think? True, that's not going to happen very often, plus it's impossible to flop either a Straight or Flush with 7-2o, so it's pretty fair to say 7-2o sucks as a starting hand. But, we do know that 7-2o can win - sometimes - so what kind of pot odds are needed to make betting on it profitable? I'll spare you the suspense; it's much higher than the 9 to 1 odds that I'm talking about here - more like 20 to 1 against four other hands that your opponents might limp with. At least now you know.
This program evaluates the "equity" a hand has in the pot. You see, there are two ways to measure the strength of a hand, preflop: pot equity and percentage of wins. They're actually quite close to one another. For example, A-A has a pot equity of 85.2% and a win percentage of 84.9% when heads-up against a random starting hand. The difference is explained by ties, where 50% of the pot is gained when the hand is tied. For practical purposes, either measurement will do, because ties don't hurt us. If you'd like a listing of hands by equity and win percentage, go here: http://www.gocee.com/poker/he_ev_hand.html/
If you watch the World Poker Tour series on the Travel Channel, you might have heard Mike Sexton refer to Q-7 offsuit as the "computer hand". That's because Q-7o is the average hand one gets as their pocket cards. In this context, "average" means that a Q-7 will win 50% of all the hands played heads up versus any random hand your opponent may hold. Just to keep the record straight, that's winning percentage; the hand that produces a 50% pot equity return is J-5 suited. But I'm going to use Q-7o as the "average" hand in my presentation to you here, because if it's good enough for Mike Sexton, it's good enough for me. I'll go into that more in a bit.
The primary information I want to convey to you is the probability of various hands winning when they're playing against four other hands - not random hands, but hands that players might limp with. For example, you might have 7-2o in the Small Blind, but it's not likely that other player chose to limp in with that hand. Most likely, you'll be up against hands like A-Ko, 3-3, Q-10s and so forth. In the case of A-Ko, someone may be trying to get "cute" by slow-playing it. In most cases, the other hands aren't worth raising in early position, but people will limp in with them. Now, 7-2o versus Q-10s is a loser, let's face it. But so is drawing to an inside Straight a loser, unless you're getting pot odds that make it a positive expectation play. As you saw in Lesson 2, an inside Straight draw on the flop (with two cards yet to come) converts to a Straight only 16.5% of the time. So, unless the pot odds are giving you at least a 6x return for your bet, you're in a negative expectation (-ev) situation should you continue to play the hand. Naturally, you might hit the inside Straight if you play it, regardless of the pot odds, but over a long period of time it's costing you $$$ to make that call. And "long-term" is the only way to approach this game.
That being the case, there are some "long-term" situations that should be called when you have huge pot odds available to your SB bet. By "huge", I basically mean the 9 to 1 odds that you'll get with 4 limpers into a pot. In the SB portion of my Limit Hold 'em Basic Strategy Matrix, there's a category of hands called "Complete Only". These are hands where you should invest only the additional 50% of the SB bet; in other words, fold them to a raise. In the case of an Ace hand, the minimum is A-8 offsuit and the math behind it assumes an "average" pot. But what if there's no raise and four limpers have entered the pot - which is now above average - would it be worthwhile to complete with a hand like A-6o, for example? If we are receiving 9 to 1 odds on our bet, we need a 9 to 1 shot at winning the hand to break even, which is a 10% probability.
Now all we need to do is compare some of the hands that are lower than the "Complete Only" category with a sampling of limping-type hands to see if any of them have a 10% pot equity on a pre-flop basis. Naturally we have to do this on a pre-flop basis, because once the 3 flop cards hit the felt, our percentages have changed and you might have to check and fold, or you might want to raise; it all depends upon how hard the flop hit you. However, on average (I hate to use that expression here, but it applies), if your hand has an equity of 10% or more, it's worthwhile to complete the blind. To continue with the example above, I compared the equity of A-6o against A-Ko, A-7s, 3-3 and Q-10s, all fairly typical "limping" hands your opponents might hold. The calculations produced by Poker Stove show that an A-6o has equity of 9% in that situation.
Just so you know exactly what that means, if you were to go all-in against the other four hands and they went all-in as well (admittedly a very unlikely situation) and if all five hands were played out to the river, you'd win 9% of the time. Well, that's not 10%, so my chart will show you that completing the SB bet with A-6 is a bad bet. But, with the propensity of players to call with Ace-anything in low-limit games, you also now know that you'll be up against such hands fairly often when you're hopefully holding one of the SB hands I do recommend you complete with, as is shown in my Limit Hold 'em Basic Strategy Matrix. As a verification, the lowest Complete Only Ace-whatever offsuit hand is A-8o and Poker Stove shows that with an equity of exactly 10%. Now, understand that rounding and what cards of which suit are in the other hands can affect the equity percentage, so what you'll see is really an approximation, but it's close. If you want to tighten up your play, then go with hands that have an equity of 11 or 12%. On the other hand, playing hands with an equity of 8 or 9% might be worthwhile, if you're really good at playing after the flop. Very likely, you'll need to be a good bluffer in such situations, but some people are, so go for it if it works for you.
Before we get into the chart, let me explain exactly what you'll see. It's on an Excel worksheet, so if you have any problems reading it here, email me and I'll send you a copy. In the left column, you'll see four hands that represent the holdings of your opponents - the limpers. In the top row, I put in a series of hands you might hold in the SB and they're hands that are "outside" the Complete Only hands shown in my Limit Hold 'em Basic Strategy Matrix. The percentage equity of the SB hand is shown in the row labeled "%". The equity of the other hands is shown below that and, of course, they add up to 100%. So, for your hand of A-6o in the SB, the equity is 9%; the equity for the player with A-Ko is 22%, it's 15% for A-7s, 21% for 3-3 and 33% for Q-10. Please remember that this is pre-flop equity, so don't get confused when people talk about hands like A-Ko being a "coin-flip" against a pair like 3-3, because that's only when those two hands are heads up against each other. Also, don't forget that you can toss this chart away once the flop comes, because then a different set of equity percentages will apply.
(Source: GameMaster)
