Hand Range and Hand Probability

Humor me for a moment while I lecture in an area where I am perfectly unqualified to say a single authoritative word. When considering hand ranges, it is facile to make simple calculations using basic information: possible hands and the number of combinations that make up those hands.

Of course, hand ranges are not static. One's picture of a villain's hand range should evolve with each action that is taken in a given hand. The hand range PF is narrowed by the action on the flop, and the hand range on the flop is further narrowed by the action on the turn.

This all seems straightforward enough. But when considering hand ranges in terms of hand combinations, it is easy to make the mistake of assigning equal probability to all hands in the range. This is my cardinal sin, and this entire thought process brought on by a hand that I mangled yesterday (details later).

The problem is one of oversimplification, and also of blind optimism. Here's an example:

Say you hold AKo, raise PF, get one TAG caller, and the flop comes K-8-5 rainbow. You bet and get raised. You assign a hand range of AK/KQ/88/55/67 and some small amount of air. You call and the turn comes a 2. You check and the villain bets 4/5 of the pot. Now, the hand range that you assigned hasn't changed much, but the hand probabilities have. That's because it isn't equally likely that he fires a big turn bet with each of the hands in his range. A lot of players would check back 67 and take a free river card. Same goes for KQ, for pot control reasons. So now the probability of your opponent holding those hands goes down. In other words, things just got worse for you, because the two hands that you beat just dropped off in probability. Even though there are far fewer combinations of 55 and 88 in his range (6 total), the probability that the villain is holding that hand just went up, since he makes that big turn bet a high % of the time with those hands.

Now on to the hand in question:

Full Tilt Poker, $0.25/$0.50 NL Hold'em Cash Game, 6 Players
LeggoPoker.com - Hand History Converter
SB: $29.50
BB: $104.75
UTG: $50
MP: $64.70
Hero (CO): $50.70
BTN: $63.20
Pre-Flop: K J dealt to Hero (CO)
2 folds, Hero raises to $1.75, BTN calls $1.75, 2 folds

Flop: ($4.25) 2 T J (2 Players)
Hero bets $2.40, BTN raises to $6.30, Hero calls $3.90

Here we can make our first hand range asssessment. It could be 22/TT/JJ, although we can discount TT and JJ since we'll often see a 3-bet from the button with those hands. It could be JT/AJ/KJ. With no flush draws, the possible semi-bluff drawing hands are 89 and KQ.

Turn: ($16.85) 5 (2 Players)
Hero checks, BTN bets $12, Hero calls $12

Turn is a blank. Villain bets 4/5 pot. At this point there's still a chance that my opponent is semi-bluffing an OESD, but that chance should be much lower now, since opponents will often take a free card and try to hit their straight. Also, QJ/KJ/AJ become much less likely, as those hands will look to pot control here, fearing that I am holding an overpair.

The key in analyzing this spot is to apply probabilities to the hands in my opponent's range. Sure, AJ is in my opponent's range, but he's raising the flop AND betting the turn no more than 15-20% of the time. With 89/KQ, he may only be continuing the semi-bluff 30-40% of the time. But with 22/JT, he's firing a 2nd barrel very frequently, between 80-100%. So even though there are more combinations of the hands that I beat and are drawing, when you apply probabilities, I am way behind the hands that he likely holds.

But ever hopeful, I call.

River: ($40.85) 3 (2 Players)
Hero checks, BTN bets $20, Hero calls $20

River is another blank, putting a spade flush up. I check and the villain bets $20, half the pot. Here, thinking about how many ways my opponent has 89 and KQ (praying that my opponent has fired a 3rd barrel with either air or a missed draw), I make the clown call. Had I properly assessed the hand range AND probabilities, it would have been an easy fold. On the river, and on the turn.

Results: $80.85 Pot ($3 Rake)
Hero mucked K J (a pair of Jacks) and LOST (-$40.05 NET)
BTN showed 2 2 (three of a kind, Twos) and WON $77.85 (+$37.80 NET)