Poker Articles

Math is always a big part of my game, and I'd like to share the thought process that we math nerds tend use for those of you who rely more on the feel of betting strength to display your poker prowess.

I've always felt that winning players, those who take advanced analytical approaches on how to bet and how to place their opponents on a hand, typically fall into two distinct personalities. There are mathematicians and there are psychologists. Each school of thought is intricate in its own way, and although the paths of analysis are often very different, both approaches lead to wise (and oftentimes almost identical) decisions.

The mathematician has a deep understanding of odds, how often people are likely to hit draws, and when it is profitable to stay in a hand as an underdog because the pot odds or implied odds allow it. He classifies his opponents into ranges from tight to aggressive based on how often they raise preflop in early or late position and by what percentage of the time they bet postflop. He calculates his chances of winning hands based on the likelihood of being able to bet someone out of a hand depending on the cards that fall and on the likelihood of his hand holding up against the range of hole cards on which he places his opponents.

The psychologist has a deep understanding of the mind, knows why people are staying in hands, and deduces when it is profitable to stay in as an underdog because he senses weakness. He classifies his opponents into ranges from tight to aggressive based on how much, how little, or how often they bet, and by how much strength they show postflop. He knows how to induce bets from his opponents when he has a strong hand and how to gain knowledge from his opponents by how they counter his own bets.

In this particular tournament, I, the mathematician, found myself with an above-average stack of 9300 chips.

Blinds were starting to increase fast (currently at 150/300), and I had been unsuccessful so far in turning up the aggression to build my stack when I won the following hand.

In the big blind, I was dealt QJ. Everyone folded to a tight player in middle position who had 11800 chips. He raised the bet to 900, and everyone folded around to me.

My opponent’s bet was only 3x the big blind, leaving me 600 to call, but this guy hadn’t raised very often in middle position. I immediately eliminated two types of hands that he could have. With a small pair, he was the type of player likely to limp with the hope of catching a set, and with a Broadway hand (KQ, KJ, or QJ), he most likely would have folded, and the fact that I myself held QJ, made this possibility even less likely.

As I stared at my QJ, I ran through his possible hole cards based on how I had seen him play previously, and determined the chances he held any particular hand were roughly as follows:

Large Ace (AK, AQ, or AJ) - 50%
Large Pair (AA - TT) - 18%
Mid Suited Connector (TJ - 67) - 12%
Weak Ace (AT - A7) - 10%
Broadway, Middle to Low Pair, Trash - Less than 10%

I calculated that if I saw the flop on this hand, I would probably be doing so as an underdog (seeing how 85% of the hands I put him on had me beat), but I decided that depending on the flop, I might be able to take the hand away from him anyway. This was player who was willing to fold 90% of the time if you bet into him when he didn’t make his hand. I decided to call.

The flop came KQ9.

I liked the flop, which gave me middle pair, a flush draw, and a gut shot, but it was by no means a made hand. I still believed if the flop hadn’t been kind to him he would fold, so I led out with a bet of 600. He immediately raised me to 1800. At this point I knew that he had either made a hand or had flopped a monster draw. It was unlikely he would raise me with a better flush draw unless he had also paired the board, but I had the Q (the only card he could pair with and still have a flush draw), so I started to feel safe about my clubs. I recalculated the hands he would have raised me with and grouped them with my preflop assumptions to come up with the following possibilities for his hole cards:

AK - 40%
KK - 15%
TJ - 10%
99 - 10%
AA - 10%
QQ - 5%
AT, A8, A7 - Less than 5%
KQ - Less than 5%
AX, Trash - Less than 5%

From these assumptions, I concluded that I was still beat but had several outs. Any club and any T was a true out in 95% of the scenarios (as long as he didn't have the A, all my clubs were good because no club could fall that would pair the board), a Q was a true out in 60% of the scenarios, and a J was a true out in 55% of the scenarios.

I had 17 semi-outs, 12 of which (any club, any T) were sure winners, 2 more of which (any Q) were likely winners, and 3 remaining of which (any J) were probably at least profitable bluffing hands even if they didn’t give me the best hand (a jack on the board, giving me two pair, would give my opponent’s possible trips a scare, allowing me to bluff having a T for the straight). There were 4200 chips in the pot, and I was looking at calling 1200 more chips of my remaining 7800. Hitting my hand would almost certainly bring even more chips to the center, and so both the pot odds and the implied odds dictated that I call.

The turn came the T, making me the straight.

The only real hand I had to worry about at this point was AJ, a hand I had eliminated from my opponent’s possible cards because of his raise on the flop. His possible trips posed a minimal threat of pairing the board (close to a 20% chance with one card to come), and even less of a threat was his chance of catching the flush and actually having one that would beat the flush that the same card would give me.

At the same time, I knew that the T was a scare card for him, and if I had made it to this point with some sort of unmade hand, it would be the perfect time to attempt a steal. The problem was, I had a hand that I could make more money with, and I didn’t want him to fold. Let him draw for a board pair when I’m an 80% favorite. I knew my opponent was likely to think I was on a flush draw and was at least scared that I held a jack, so I wanted to bet an amount that would give him a little less than 5:1 odds to catch his boat if he was holding trips.

There was 5400 in the pot, and I had 6600 chips left. Pushing here would not only make him believe I had the straight, but also, it would make him call 6600 chips for a 20% chance of winning 12000, giving him less than 2:1 odds for the call - way too much. I did some quick math, and determined that a bet of 1800 chips would put 7200 in the pot, giving him 4:1 to catch his boat. I was half hoping for a reraise, but he just called.

There were now 9000 chips in the pot. My stack was left with 4800 chips, and my opponent was sitting on 7300. His call on the turn gave me even more information about his hand. I couldn't fathom the possibility that this tight player was still holding on to AK with 4 to the straight sitting on the board. There was a small chance that he was still holding on to AA as someone who just couldn't let it go, but I was pretty confident he was sitting on trips. Again, I recalculated what I thought his hole cards might be:

KK - 30%
99 - 30%
QQ - 20%
TJ - 13%
AA - 5%
KQ - Less than 5%
AX, Trash - Less than 5%

From this knowledge (considering an 80% chance he was sitting on trips), I decided that if the board paired, I would shut down, humbly leaving the hand with my remaining 4800 in chips. With any other card (including any club that hit), I would strike for more chips.

The river came the 6, a complete blank.

Now I had a decision to make. How much would he call? I figured if he was 80% sure I held a jack, he'd be willing to call an amount that gave him better than 5:1 odds. With 9000 in the pot, a bet of 2000 would give him 5.5:1 to make the call.

I sent out my bet of 2K, but the guy had an impeccable read on me. After a few minutes of deliberation, he folded his hand, showing pocket kings and leaving himself with 7300 chips instead of a crippled 5300.

Nonetheless, I was happy with the boost that math had given to my stack.