How Much Does Math Matter?
This is the third of three columns that I have audaciously titled “Poker Made Simple.” In truth, I think that this series demonstrates that poker is simple on the surface and, underneath the still waters, very complex. The basic question is always simple: “Should I raise, call, bet, or fold?” The questions you might consider to answer this simple question are very numerous and difficult.
In the first installment in this series I explained a simple truth about poker: math defines the most profitable play you can make in a given situation if you know all the variables. In the second part I showed that the only variables that really matter are: what’s he got and what will he do with it? You know your cards and you know the board cards. The only things you don’t know are your opponents’ cards and how they will play them. Any question you may ask yourself (e.g. Is my opponent drunk?) is simply an angle of attack on what you really want to know: what’s he got and what will he do with it? Good poker players are able to accurately put their opponent on a hand, understand how the opponent will play, and understand the math of the situation in which they find themselves.
Let’s look at a couple of examples that will illustrate how much math matters. Note that I have somewhat simplified the following examples – in part to make it easier to understand the point and in part because a thorough mathematical exam is beyond the scope of this article (not to mention my abilities).
Check, Bet, Call, Raise, or Fold?
The game is $10-20 limit hold’em. You and one lone opponent remain. The pot is $95. You have 5c-4c and the board is Kc-Qc-8h-7h.. Your opponent has been betting into you the whole way. You have played with him frequently and you know with some certainty that when this opponent bets he has at least top pair which he will not fold unless the board gets really dicey. In other words, you have defined the variables you are interested in: what’s he got (at least top pair) and what will he do with it (bet and call if you raise). You can rule out the possibility that he has a bigger flush draw – no hand that is a bigger flush draw would have been “at least top pair” on the flop. So, you know with near certainty that you have at least 10 outs and probably 12 outs. If your opponent already has three-of-a-kind then the 8§ or 7§ are no good to you because either card will make your opponent a full house or quads – you have ten outs (the remaining clubs or any six). If your opponent has A-K then you will have 12 outs (any club or any six). What is the correct play?
Let’s look at each option and the mathematical consequences. For simplicity we will assume the more modest assumption that you have ten outs:
Folding. Ignoring what you have lost so far on the hand, if you fold you will neither win money, nor lose money. Your total is $0.
Calling. Your opponent bets the turn. You must call $20 in order to have a chance at winning $115 (the original $95 in the pot plus his bet of $20). With 10 outs you will make your hand 22 percent of the time (10 out of the 46 cards). If you play 100 hands then 78 times you will lose (call $20 on the turn and then fold the river when you miss). Your total loss will be $1560. You will win the other 22 hands – for $135 per hand (the original $95 plus $20 from your opponent on the turn and $20 on the river – it could be $40 on the river, but we will again make the more modest assumption). Your net for the 22 hands you win be $2970. Finally your net profit over 100 hands will be $1410 or $14.10 per hand.
Raising. Your opponent bets the turn. This time instead of calling $20 you raise to $40. He will call (he rarely folds top pair). You still have ten outs and you still will make your hand 22 percent of the time. Once again, if you play 100 hands then 78 times you will lose. Your total loss will be $3120. You will win the other 22 hands – for $155 per hand (original $95 plus $40 on the turn and $20 on the river). You net for the 22 winning hands will be $3410. Finally your net profit will be plus $290 or $2.90 per hand.
Obviously, your best option is to call and then bet or raise on the river. On the turn, folding is break even, raising is slightly profitable, and calling is the most profitable. This example illustrates what I preach as the basic low-limit poker strategy: bet with the best, good draw to invest, fold all the rest. At low-limit poker you should generally bet or raise when you have reason to believe you have the best hand and call when you have a good draw.
A More Interesting Example
Is it that simple though? Is it enough to understand basic pot odds? Well, it depends. If you want to play profitably at low and mid-limit poker a rudimentary understanding of pot odds is all that is required. You can, however, go to a much higher level of understanding.
It is a no-limit hold’em tournament. The blinds are $50 and $100. You are the big-blind. It is folded all the way around to the small- blind who makes it $300 to go. You started the hand with $1000 (you have $900 left after posting your blind). You look at your cards and see 7-2 offsuit. Obvious fold right? Wait. Not so fast! Pause for a second. Give it some thought. What are the variables? What’s he got and what will he do with it? Suppose, for example that you have a good handle on this opponent (you have him in a box) and you are reasonably certain that he will raise in this situation with any ace, any pair, and any two cards that are both ten or better. Also, you are also reasonably certain that he will only call a re-raise to $1000 with A-A, K-K, Q-Q, J-J, A-K, or A-Q.
Suppose you re-raise all-in. When you do, he will fold 85 percent of the time. If you play 100 hands you will win 85 without seeing a flop for a total profit of $34,000 (his $300 plus your big-blind of $100 85 times). If he calls then you know he has the smaller range of hands listed above. Against this smaller range of hands 7-2 offsuit will win just 23% of the time. So, of the remaining 15 hands you will win about 3.5 of them (profit of $3,850) and lose 11.5 of them (loss of $10,350). Thus, in the situation described if you re-raise all-in you will net $27,500 or $275 per hand.
Surprised? Note that I have not considered the considerably more complicated option of simply calling his $300 bet pre-flop. You get into a big variety of situations if you start working out the possibilities post-flop. Nor have I considered the consequences of being knocked out of the tournament (which you will be 11.5 percent of the time that you try this play).
The point, however, is that once you are good at putting your opponents on hands and knowing what they will do with them, a thorough understanding of the math of the situation you find yourself in will show you the most profitable path. Even 7-2 offsuit is a profitable hand in the right situation. Believe me when I tell you that every single world class player has, at the very least, a strong intuitive feeling for the math of any given situation. He may not have thought of this specific situation, but he has thought of one quite similar and he has a great depth of understanding to draw upon in making the decision this time.
I do not pretend that this series will, by itself, make you into a great poker player. Indeed, only you can make you into a great poker player with sufficient study and practice. What this series of articles does do is show you one way of approaching your study. When I am asked a poker question I first pause to consider what I am being asked. Is it a math question or a psychology question? Is it about defining the variables (what’s he got and what will he do with it?) or is it about the most profitable course to take if you know the variables?