Texas Holdem Probabilities - Flops

By Tony Guerrera

This is part of a series of articles covering important probabilities in Texas hold'em. The second and third articles in this series covered probabilities associated with hitting hands–the typical stuff you’d expect in articles about poker probabilities. The first article covered preflop fold equity and showed how poker math goes well beyond hitting hands.

Like the first article, this article has nothing to do with you hitting hands. Instead, it focuses on how often your opponents hit hands.

Get ready to learn about Texas holdem probabilities on the flop.

What’s So Special About Two-Thirds?

Suppose the flop is Q83. You didn’t hit it. If you’re heads-up on the flop, what’s the probability that your opponent didn’t hit it either? To be absolutely rigorous, we should put your opponent on a hand distribution. But as a starting point, let’s just assume that your opponent is on a random hand. You have two cards, and we know that the board hit neither of them; the deck effectively has 47 cards. The probability that neither of your opponent’s hole cards matches a board card is .

The probability that he completely missed the flop is just under two-thirds. We haven’t accounted for pocket pairs and flopped draws, but you’re in good shape if your opponent is playing straightforward hit-to-win poker.

Two-thirds is why continuation betting is so powerful. It’s why you should show relentless aggression when you’re heads-up postflop against tight-passive opponents (unless, of course, you’re playing an ultra tight player whose likely distribution consists largely of AA and KK). And it’s why tactful aggression reigns supreme in the shorthanded setting.

Since you won’t always be heads-up postflop:

Probability that an Unpaired Flop Won’t Match Any of Your Opponents’ Hole Cards if They Have Random Hands

This table suggests that betting on the flop can be profitable regardless of your cards if you’re betting into one or two relatively straightforward opponents. And if you have position on three or four straightforward opponents who check to you on the flop in a hand that was unraised preflop, they are no longer on random hands. Their checks on the flop imply weakness, and you should consider betting. (If you raised preflop and ended up with three or four callers, the story is a bit different–checks no longer give you information because even straightforward opponents will often look to check-call or check-raise a preflop raiser).

Paired Flops

When the flop is paired, players tend to get scared. They worry about the possibility of getting raised by someone with trips. But is getting raised by someone with trips on a paired board any worse than getting called by someone with a pair on an unpaired board? Of course not; these results are exactly the same. The numbers show that it’s even less likely that your opponents have hit the board on a paired flop: 

Probability that a Paired Flop Won’t Match Any of Your Opponents’ Hole Cards if They Have Random Hands

Beyond the Numbers

You shouldn’t mindlessly bet shorthanded flops 100% of the time just because of the numbers in this article. If you abuse these numbers too much, even your most timid foes will begin playing back at you. And that’s bad: if your opponents are folding too much and not bluffing enough, getting them to play back more would be getting them to correct themselves in the direction of optimal play. Additionally, the numbers in these tables assume that your opponents are on random hands. Suppose you raise with 87s, a very tight player calls–someone whose call signals a distribution looking something like {AA-JJ, AK-AQ}–and the flop is AKQ. Continuation betting in this situation is pretty much suicide.

Besides knowing the results of rough calculations, successful thievery at the tables involves considering things such as flop textures, table trends, your opponents’ hand distributions, and the likelihood of facing a rebluff. And you need to be willing to change gears. For example, you see a three-handed flop, and you’re first to act. Lead bluffing on the flop has been successful, but since you’ve been going to that well somewhat excessively, you mix things up by checking the flop with the intention of lead bluffing the turn if both of your opponents check behind and the turn card is not likely to have hit your opponents’ hands.

By combining knowledge of the numbers with knowledge of the game’s subtle nuances, you should be able to implement a well-balanced strategy that will keep your opponents off guard. Stay on your toes, be creative, and may many successfully stolen pots come your way!