Texas Holdem Probabilities - Pocket Pairs

By Tony Guerrera

This is part of a series of articles covering important probabilities in Texas holdem. The math of Texas holdem extends well beyond the probabilities of hitting hands, but knowing the math associated with hitting hands is vital. This article covers Texas holdem probabilities related to hitting hands with pocket pairs.

Texas Holdem Probabilities - Pocket Pairs Preflop Considerations

When you have a pocket pair, life is usually good when you flop a set or better. Sure, AKQ can be a tough flop to play with QQ against a somewhat tight opponent who 3-bet you preflop. But exceptions like these are why I covered myself saying “usually!”

We can’t cover all the exceptions, but we should address flopping a set or better on one particular class of flops: tripped flops. A tripped flop is one where all three board cards are the same rank. Sometimes, boats made with tripped flops aren’t problematic. Other times, they are troublesome. With that in mind, let’s do the math two ways:

Excluding Boats Made from Tripped Flops

Including Boats Made from Tripped Flops

Fortunately, tripped flops don’t contribute significantly towards your probability of hitting a set or better: the numbers are close enough such that you only need to remember one number instead of two. The number I suggest keeping in mind is 7.5:1. 

If you’re playing your small pocket pairs purely for sets, you’ll only profit in the long run if you’re getting at least 7.5:1 on your preflop call after accounting for pot odds (the money currently in the pot), implied odds (money you anticipate your opponents contributing to pots that you win), and reverse implied odds (money you anticipate contributing to pots that you lose–like when you hit a set that doesn’t improve against an opponent’s flush). You won’t get action every time you hit a set, and even if you do get action, it won’t necessarily be big action…especially in today’s playing climate where many players seem unwilling to lose lots of big blinds with top pair/top kicker. Accurate player profiling is the key to assessing your odds properly.

If you’re not getting proper set odds, you need to dump your pocket pair unless you have lines of play that win you the hand when you don’t flop a set or better. With high pocket pairs, the chances of you flopping an overpair are quite good. And though overpairs can be vulnerable, you should be able to play them profitably regardless of whether you flop a set–extracting value when you’re ahead and muckalucking when you’re behind.

The value of an overpair is a function of the flop and your opponents’ hand distributions. A good launching pad for some initial number crunching is to make the following two assumptions:

1.)Flops containing three consecutive cards are unfavorable unless you have a straight draw (for example, 765 and 654 are favorable flops for 88).

2.)Monotone flops (three cards of the same suit) are unfavorable unless you have AA, where one of the aces is the same suit as the board suit.

The following table, taken from p. 112 of Killer Poker By The Numbers (Lyle Stuart, 2007), gives the probabilities of and odds against a favorable flop for each pocket pair:

Table 1: Probability of Getting a Favorable Flop with a Pocket Pair 

Table should give you a rough idea of a basic strategy for playing pocket pairs, but it should be far from being your only consideration. It’s easy to become comfortable with playing medium and low pocket pairs solely for set value, but pocket pairs can have good showdown value in heads-up or three-way pots when they aren’t overpairs. Additionally, you don’t always have to showdown the best hand to win. Remember that you can also win hands by making your opponents fold–it doesn’t matter whose cards are better if you don’t have to show your cards. Ultimately, the numbers combined with your knowledge of how your opponents play will dictate the best way to play whatever situations you get into.

Postflop Considerations

If you have a pocket pair, and you’ve seen the flop, two scenarios exist for improving your hand: 1.) You didn’t flop a set or better, meaning that you’re drawing to a set 2.) You flopped a set or better, meaning that you’re drawing to a boat or quads. I’m not saying that you necessarily need to improve to win. I’m just saying that these are the two scenarios for improving, so we can do the math.

First, let’s consider the probabilities involved when you don’t flop a set. If you don’t have at least a set on the turn, the probability that you’ll hit a set on the river is .

The odds against you improving are 22:1. If you don’t have at least a set on the flop, the probability of improving will be a function of whether you’ll get to see the turn and the river, or the turn only; you need to predict whether your opponents will bet the turn in a fashion that will shut you out of the pot. If you’ll only see the turn, then the probability of you hitting a set is .

The odds against you improving are 45:2 = 22.5:1. Meanwhile, if you’re confident that you’ll get to see the turn and the river, the probability that you’ll hit one of your two outs is  (notice how this is close to–but not exactly equal to–twice the probability of hitting with one card to come).

The odds against you hitting at least one of your two outs with two cards to come are .

Unless your opponents min bet or yield huge implied odds on the turn or the river, you probably won’t be able to withstand a bet with a pocket pair that needs to improve to a set on the turn or the river in order to win. Your unimproved pocket pair either needs to compare favorably to your opponents hand distributions, or you need a line of play that can get opponents with superior hands to fold.

Second, we get to the good part: probabilities associated with improving your hand when you have a set. If you have a set on the turn, you typically have 10 outs. 9 outs give you a full house, and 1 out gives you quads. Of course, these numbers change in specific situations, like if an opponent has a higher set. But let’s keep things simple and assume that you have 10 outs. With 10 outs, the probability that you’ll get a full house or quads on the river is .

The odds against improving your set are 18:5 = 3.6:1. You might not always get these odds–especially if opponents with flushes become timid when boards pair. But the numbers suggest that at least some times will exist where you can justify drawing to improve a set that’s beaten on the turn.

Now, let’s consider the probability of getting a full house or quads if you flop a set. Like all draws on the flop, you need to consider whether you’ll only see the turn, or if you’ll get to see both the turn and the river. Granted, sets are typically huge hands that you’ll go the distance with, but times exist when you simply know that you’re beaten. For example, you have TT, and a very timid player goes nuts when the flop is 89T–you have top set, but you’re certain that he has QJ for the straight. When these situations arise, you need to dissociate yourself from being excited about having a set; these situations require dispassionate thinking and calculating.

If you’ll only get to see the turn, the probability that your set will improve is .

The odds against you improving are .

If you get to see the turn and the river, life is much better. The probability that you won’t improve is .

Therefore, the probability that you will improve is .

The odds against getting a full house or quads with two cards to come are .

This 1.99:1 is why sets are so powerful! Most of the time, your set will be miles ahead of your opponents, and even though you’re already ahead, you improve about one-third of the time!

Final Thoughts

This article discussed the probabilities associated with hitting hands in Texas hold’em when you have a pocket pair. The odds against flopping a set or better are about 7.5:1, and table 1 provides an excellent starting point for developing a strategy that also incorporates flopping overpairs on benign boards. Heading to postflop play, we saw that the probability of improving to a set postflop is extremely small. However, if you flop a set, you’ll get a full house or quads roughly one-third of the time if you get to see the turn and the river. I would include a table of all the probabilities and odds associated with postflop play when you just have 1 card to come, but calculating the odds on the spot with 1 card to come should be a natural part of your game: just take the ratio of cards that miss to the cards that hit.

Though we went in-depth about the math involving hitting hands with pocket pairs, realize that to compete in tough games, you need lines of play that extend beyond flopping sets and overpairs. The numbers in this article are important, but don’t forget the bigger picture and the balanced overall game plan that you must bring to the table.