My new book, Texas Hold'em Odds and Probabilities: Limit, No-Limit, and Tournament Strategies, was released recently, and I think it will help a lot of players in developing a good mathematical understanding of the game. The book does more than just show you how to calculate the odds; more importantly, the focus is on how to apply odds to make better decisions.

The book includes an analysis of an interesting hand that came up in the 2004 Borgata Open between Josh Arieh and Phil Ivey that I thought would be interesting to discuss in this column.

The blinds were $12,000-$24,000 with $3,000 antes. A player in middle position raised to $80,000. Arieh called with 3-3 and Ivey moved all in for $403,000. The original raiser folded and it was up to Josh, who had Phil out-chipped. There was $581,000 in the pot. What would you do?

When watching the broadcast, my initial reaction was that Josh should fold. He was either in a coin-flip situation against two overcards or a big underdog against a big pocket pair. But he also had pot odds on his side. The question is whether or not they were enough.

Josh is faced with a $323,000 call to win $581,000, giving him 1.8-to-1 pot odds. What are the various hands that Phil could be holding? I have never played Phil Ivey, so I only have information about his style that I have gathered from watching him on TV. The player who initially raised had enough chips remaining that Phil probably thought he could get him to fold to an all-in reraise unless his opponent was sitting on a premium hand. Given that Josh had called, Phil was probably thinking that Josh was weak and would fold his hand. Therefore, Phil's main concern was the original raiser.

Phil had a below-average stack and needed to make a move, so this offered a good spot. However, I believe he also respected the original raiser, so he would not make this play without a quality hand. The range of hands with which I suspect Phil might move in here includes all pairs 10-10 and higher, A-K, and A-Q.

This is where we can use probabilities to help us make our decision on whether calling is profitable or not. We first need to determine the probability that Phil is holding a pocket pair. Each pair has six combinations (Aspade Aheart, Aspade Adiamond, Aspade Aclub, Aheart Adiamond, Aheart Aclub, Adiamond Aclub). There are 16 different card combinations when holding A-K and 16 combinations of A-Q. That gives Phil a total of 62 card combinations, of which 30 are pairs.

This type of analysis may seem a little complex during the heat of battle. My book includes a chart that shows the probability of an opponent holding a pocket pair given common ranges of hands. Based on the range of hands we put Phil on, the probability that he is holding a pair is .48 (30/62). To make things simple, let's assume that Phil holds an overpair 50 percent of the time and two overcards the other 50 percent of the time.

A pair of threes will win about 18.5 percent of the time against an overpair and about 53 percent of the time against two overcards. We can now calculate the expectation of this bet.

Against an overpair: (.185 × $581,000) - (.815 × $323,000) = -$156,000

Against two overcards: (.53 × $581,000) - (.47 × $323,000) = $156,000

Given that we estimated an even chance of holding an overpair or two overcards, it turns out that calling is a break-even proposition. Now we know what Mike Sexton meant during his broadcast when he said that Josh was faced with a tough decision! Since we have a break-even play here, the question now becomes, should you risk doubling up Phil Ivey to try to knock him out now? Phil was acting right behind Josh, and I'm sure that Josh would try anything to get Phil out to improve his chances. Josh decided to call, and Phil was knocked out of the tournament with A-Q when he got no help from the board.

Realize that the odds play just one role in the decision-making process. The first step is to determine the variables, which in this case is the range of hands with which you think Phil will go all in here. Estimating this range is a process of using your prior experience in playing against your opponent, your opponent's current playing style and mental state, and any reads or instincts that might make you feel a certain way. Once you determine a range, you can do some simple math to determine if you are getting the correct pot odds to call.

Note that if Josh thought Phil might raise with a few more pairs than in my analysis, he would have a negative expectation on this call. If he thought he might raise with hands like A-J or K-Q, the expectation would be slightly positive. In either case, the decision was quite close to a break-even proposition.

May the odds be with you! spade

Matthew Hilger is the author of Internet Texas Hold'em and Texas Hold'em Odds and Probabilities. You can ask Matthew questions or follow the first Internet Player of the Year race at Matthew's website, http://www.internettexasholdem.com/.