Texas Holdem Probabilities - Unpaired Cards Preflop

By Tony Guerrera

The math of Texas hold’em extends well beyond the probabilities associated with hitting various types of hands, but the math associated with hitting hands should be a fundamental part of your poker knowledge.

In this article, we’ll talk about the preflop math of unpaired hole cards. This article is mainly about the results.

One-Pair, Two-Pair, Trips, Boats, and Quads

Assume you hold two unpaired hole cards. For now, ignore suitedness and connectivity–meaning that we’re not concerning ourselves with straights or flushes. The probability that you’ll miss the flop (none of the board cards match your hole cards) is .

The probability that you’ll hit the flop (meaning that at least one board card matches your hole cards) is .

If you’re playing short-handed against opponents who bet flops pretty much 100% of the time, you’re potentially in trouble if you’re playing strict hit-to-win poker. I say “potentially,” because maybe your opponents have huge holes in their games–like yielding gigantic implied odds that compensate for all the small pots they steal with their perpetual flop bets. But most likely, you’ll need lines of play beyond strict hit-to-win poker to show a long-term profit. Let’s break things down further by examining the probabilities of flopping certain classes of hands with unpaired hole cards.

Table 1: Probabilities of Flopping Specific Hands with Unpaired Hole Cards

Hand Type	Probability
One-Pair
Two-Pair
Three-of-a-Kind
Full House
Quads

You’ll flop a pair about 28.96% of the time. Meanwhile, you’ll flop two-pair, three-of-a-kind, a full house, or quads only 3.47% of the time. With hands like 82o, playing a flopped pair can be really tricky. You’d really like to have two-pair or better with them, but the odds against such a magic flop are .

Getting 27.82:1 on your money in a limit game is damn near impossible. No-limit hold’em is a game of implied odds, but that’s not a license to be an optimist who overestimates your implied odds. When you hit two or three board cards, it’s less likely than usual that your opponents have hit the flop. And often, when you hit a big hand with junk, you don’t want tons of action because your big hand isn’t really as powerful as it seems. For example, if you flop 883 with your 82o, opponents willing to invest lots of chips will generally have you crushed. The implied odds just aren’t there–no matter how deep the stacks are. Junk hands like 82o should be instamucks if you’re playing strict hit-to-win poker. But contrary to what all the ABC pundits say, they aren’t always instamucks; any two cards are playable if you have lines of play that carry substantial fold equity.

(Having just admonished the hoards of generic poker theorists out there, I will say that you should avoid getting too fancy for your own good; creative play isn’t synonymous with profitable play, and ABC lines of play are sometimes your best option. Just don’t forget to evaluate all options when making your decisions–diamonds sometimes lurk in the rough.)

Flushes and Straights

Ignoring straights and flushes made the math easy, but it doesn’t tell the complete story. If your hole cards are suited, the probability that you’ll flop a flush is .

The probability that you’ll hit a flush if you see all five board cards is about .0640. Virtually all of those flushes are ones where exactly three board cards are of your suit, which is good since those are usually the only flushes you want.

Just under 60% of the flushes you make with five cards to come are ones where you flop two of your suit–an event that happens about 10.94% of the time. Considering that it’s easy to be shut out of pots when you only flop 1 one card of your suit, real playing conditions will dictate that more than 60% of your flushes will happen when you flop two of your suit.

These numbers aren’t huge. In fact, they seem similarly dismal when compared to the probabilities associated with hitting two pair or better. But when combining flush possibilities with those of flopping two pair or better, the numbers aren’t horrific. Adding the probability of you flopping a flush (.0084), the probability of you flopping two of your suit and hitting a flush (.0383), and the probability of flopping two pair or better (.0347), we get .0814–making the odds against this set of events about 11.29:1. This suggests that playing hit-to-win poker with any two cards may be a viable option in some playing conditions. But it certainly won’t be profitable in all playing conditions. As always, know your opponents! For example, many players shut down when three to a suit are on board, and this lack of implied odds can make it difficult to draw to flushes profitably. Generally, you’ll need lines of play not involving hit-to-win poker to play suited trash profitably.

Now, let’s throw connectivity into the mix, and calculate the probabilities of making straights involving both hole cards. Soon, we’ll see that thinking of connectivity in terms of connectors, 1-gaps, 2-gaps, and 3-gaps isn’t necessarily the most conceptually driven way to think about connectivity. I’ll introduce some new terms later, but since I think they’ll make more sense after this discussion, I’ll stick with the widely used terms for now. 

Let’s start with connectors–specifically JT-54, the ones that make the most possible straights using both hole cards. All the connectors JT-54 are mathematically equivalent when it comes to making straights, but since it’s easier to use examples, let’s use JT. For JT, the available straights are {789TJ, 89TJQ, 9TJQK, TJQKA}.

For each straight,  combinations exist for the three board cards that complete the straight, and  combinations remain for the other two board cards. Since   5-card boards are possible, the probability of getting a particular straight with 5 cards to come is . And since four types of straights are possible, the overall probability of getting a straight with 5 cards to come is . The odds against hitting a straight with five cards to come are

We could make an adjustment such that we ignore tripped boards and boards containing four or five cards of the same suit, but since these events are rare, we at least have an approximate idea of the odds involved.

The math is identical when considering straights involving 1-gaps, 2-gaps, and 3-gaps. All that changes is the number of possible straights. For hands capable of making 3 straights, 2 straights, or 1 straight, the respective straight probabilities are , , and .

Now, it’s important to note that whether a hand is a connector, a 1-gap, a 2-gap, or a 3-gap doesn’t uniquely identify how many straights a hand can make. For example, QJ and 43, hands that pretty much everyone calls connectors, can only make 3 possible straights. They appear to be connectors, but they are effectively 1-gaps. With that in mind, I’m going to define some new language to the poker lexicon–language that’s more conceptually driven when it comes to connectivity.

Another way of saying “connect” is “slap together.” Therefore, hole cards capable of making straights should be called “slappers.” I propose the following:

The slapper terminology reflects the fundamental nature of hole card connectivity, and it’s fun to use. Instead of saying “my straight won the hand,” we can now say things like “my straight slapped my opponent’s bankroll into oblivion!” Or, instead of saying “my opponent’s suited junk hit two pair and cracked my aces,” we can now say “my opponent’s suited peacekeeper killed my aces.”

With this new nomenclature in place, I present you with the following table:

Table 3: Probabilities and Odds for Straights Involving Both Hole Cards if You See All Five Board Cards

The odds against hitting a straight with five cards to come aren’t bad for four-way slappers, and when also factoring in the possibility of flopping two pair or better, four-way slappers become quite playable in the hit-to-win paradigm (though again, it never hurts to have more than hit-to-win in your toolbelt). Three-way slappers are a little bit worse, but not horrific. Meanwhile, the numbers for two-way slappers resemble those that govern hitting flushes, and the odds against hitting a straight with a one-way slapper are further compounded by the difficulty in getting opponents to cough up chips when three consecutive cards are on the board. To play unsuited two- or one-way slappers properly, you generally need more than plain hit-to-win poker.

To complete our look at unpaired hole cards preflop, we need to examine suited slappers. The counting required to run the numbers is a bit more complicated because you need to avoid double counting–overlap exists between the number of possible straights and the number of possible flushes:

P(Straight or Flush) = P(Straight) + P(Flush) – P(Straight and Flush)

I’ll show the work involved to calculate P(Straight or Flush) for suited four-way slappers, and then simply give you the results for the other suited slappers.

From earlier in this article, we already know that .

And though I quoted P(Flush) earlier as being about .640, it’s precisely .

To figure out P(Straight and Flush), consider a board where you’ve made a straight with a suited slapper. Table 4 outlines the possibilities for the specific case where you hold 78suited (clubs) and you make a straight with 456 on board.

Table 4: Combinations for Getting a Straight and a Flush with 78suited (clubs)

Board Cards	Combinations for Straight Cards	Combinations for XX	Total Combinations
456(3clubs)xx	1		1081
456(2clubs)xx (1club)			3078
456(2clubs)xx (2clubs)			324
456(1club)xx (2clubs)			1215

Thus, there are  boards that contain both straights and flushes when the board is 456. Since there are 4 total possible straights, there are therefore  total combinations that yield both straights and flushes.

There are then  boards that give you a straight, a flush, or both. The probability of getting a straight or a flush with 5 cards to come with a suited four-way slapper is .

The odds against getting a straight or a flush are . Table 5 shows the probabilities and odds associated with getting a straight or a flush with all suited slappers.

Table 5: P(Straight or Flush) for Suited Slappers

Table 5 demonstrates that suited 4-way and 3-way slappers are similar in strength, and that even 2-way and 1-way slappers can be playable in certain circumstances under the hit-to-win paradigm–especially considering that we haven’t factored in the extra .0347 probability of flopping two pair or better.

Closing Words

I've included some slightly laborious calculations in this article for those interested in the method behind the madness. After all, knowing how to do these calculations can help you with other homework you may wish to do away from the tables. However, from the pure playing standpoint, the only important thing is the results.

You’ll flop at least one of your hole cards about one-third of the time–and a majority of that one-third is when you flop exactly one-pair. If you’re playing and unsuited peacekeeper with the intention of flopping two pair or better, the 27.82:1 odds against that happening make it a losing proposition. Moving on to flushes and straights, we saw that the probability of hitting a flush, .0640, makes it hard to play hit-to-win poker if you’re solely relying on hitting flushes. And while the odds against hitting a straight with a 4-way slapper aren’t horrendous at 6.66:1, we saw that suited slappers (especially 4-way and 3-way) become extremely playable under a lot of playing conditions–even if you’re just playing strict hit-to-win poker. 

You can’t win without knowing the numbers, but you also can’t win if you don’t move beyond thinking about poker in terms of hitting hands. See all the angles, and you’ll put yourself on a path to see all the chips.