Poker Math - Part II

Poker math isn't for just the nerd faction of poker players, it's for everyone. If you can add, subtract, multiply, and divide, you can use mathematics as a weapon at the table. In my last column, I explained the everyday poker math terms "odds," "combinations," and "outs."



In this column, I want to take on "range of hands," "pot odds," and "equity," and show why they are important. I often hear the advice, "Don't put your opponent on a specific hand, but on a range of hands." It's good advice, but it's not quite enough. Let's explore this advice from a mathematical standpoint, using a hypothetical (but realistic) poker situation. You have raised with a medium pair – say, 8-8 – in no-limit hold'em, and an opponent has reraised you all in. You "know" your opponent has either A-K or a big pair – queens or better. By "knowing" this, you have assigned your opponent a range of hands. That's great, but now what do you do? You're either a big underdog (4.5-1 if your

opponent has an overpair) or a small favorite (about 1.2-1 against A-K). But that doesn't necessarily mean you should fold. You have to go a few steps further.



First, you have to determine the number of hands against which you're a small favorite or a big underdog. But, Matt, you might say, I'm a big underdog against three hands queens, kings, and aces – and a small favorite against one hand – A-K. Right? No. This is where combinations come in.



In hold'em, there are six prefl op combinations that give you pocket aces – A, A, A, A, A A, A A, A, A, A, A. You can't get aces any other way. In fact, there are six ways you can be dealt any specifi c pocket pair. There are, however, 16 ways you can be dealt any specific unpaired hand. Take A-K. You can have any of the four aces with any of the four kings. Four times four is 16. So, in the example above, there are 18 ways (six combinations for queens, six for kings, and six for aces) your opponent can have an overpair, and 16 ways he can have A-K.



OK, now you've counted correctly, and it is more likely, by a score of 18 to 16, that you are against an overpair than A-K. You're a big underdog to an overpair, and only a small favorite against A-K. Now you can fold, right? Wrong. We are missing a vitally important piece of information – namely, the size of the pot. If your opponent's raise is small enough, you should call all in even though you're likely to be a big underdog. I'll explain why.



In poker, we are often comparing the amount of money we have to call to the amount of money in the pot. The ratio of these two amounts is our pot odds. For example, with blinds of $1$2, I raise to $6, and my opponent moves all in for $21. Everyone else folds. I have to call $15 ($21 minus $6) to win $30 ($21 plus $6 plus $2 plus $1). My pot odds are 30-15, which is equivalent to 2-1. So, how often do we need to win the pot in order to call when our pot odds are 2-1? Well, every time we win, we triple our $15 investment ($15+$30=$45). So, if we lose twice in a row but win the third time, we break even. Therefore, we need to win more often than one time in three to make calling correct; we have to be a 2-1 underdog or less.



Here's another way to look at it: If you are a 2-1 underdog to win the pot, that means one time in three, or about 33 percent of the time, you will win the pot. In this situation, we poker math types like to say that you have 33 percent equity. If there is $30 in the pot and we have to call $15, we will break even if we have 33 percent equity. To prove this, let's say we call. The pot becomes $45. We are worth 33 percent of that $45, which equals $15. This is the same amount of money that we had to call in the first place. Let's say the pot had been $31 and we had to call only $14, still with the same 33 percent equity. Now if we call, the pot still becomes $45 and we are still worth $15. That's a dollar more than it cost us to call, so in the long run, we make $1 by calling. We make money because our pot odds were 31-14. We needed only 31 percent equity (14 divided by 45) to make calling correct, and we had 33 percent. If you compare your equity with the equity you need to call based on the pot odds, you can always determine whether to call an all-in raise.

Equity becomes especially useful when thinking about an opponent's range of hands. Let's get back to the pocket eights. If we convert our odds to percentages (something we covered last column), our equity against an overpair is about 19 percent, while our equity against A-K is about 54 percent. That's great, but what's even better is that we can calculate our equity against the range of hands: QQ, K-K, A-A, A-K. We determined above that there are 18 ways our opponent can have an overpair, and 16 ways he can have A-K. To get our overall equity, we just weight the different equities and add them. There are 34 combinations of hands our opponent can have (16+18=34). So, in this case, 18 times 19 percent for the overpairs, plus 16 times 54 percent for the A-K, divided by 34, gives us an overall equity of about 35 percent. (The exact number is 35.8 percent. I got that by using PokerStove, available for free at www.pokerstove.com.)



Our pocket eights will win the hand more than one time in three even if our opponent can have only queens, kings, aces, or A-K. So, if we're getting 2-1 pot odds, whereby we need only 33 percent equity, we are supposed to call. In fact, if we're getting more than 1.8-1 pot odds, we're supposed to call. Many no-limit players would fold two eights when facing an all-in reraise, even if they were getting 2-1 on their money. They are costing themselves chips.



It's very rare that an opponent's range is strong enough to make folding correct when you're getting 2-1 pot odds against an all in. Just look at this example. Our opponent's range was incredibly strong, and it was still correct to call with a measly pair of eights. In real life, few opponents have ranges as strong as the one described here. In addition, once you start calling some all-in reraises, your opponents will be much less likely to come over the top of you in future hands. Always think in terms of your equity, in terms of the value in putting chips into the pot. Don't just say, "I'm either way behind or slightly ahead, therefore I fold." ♠



Matt Matros finished third in the 2004 WPT Championship, and cashed four other times in major tournaments last year. His book, The Making of a Poker Player, is on the shelves now. Part I of this series can be found at www.CardPlayer.com.