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By Sean Carroll | July 9, 2006 2:23 am

I know the tension has been building, so without further adieu, I present the answers to our poker quiz! And you should listen to what I say, as I am a recognized expert in the field.

Remember the set-up: you’re playing Texas Hold’Em, so you have two cards to yourself, and (eventually) five cards face-up in the middle, and your hand consists of the best five cards you can choose from your two and the five community cards. Which of the following has the best chance of winning against somebody else’s (unknown, obviously) cards at a showdown?

• Jack-10 suited
• Ace-7 unsuited
• Pair of 6’s.

Note that this is not really a poker-strategy question, it’s just a math question. There is a separate issue, which is “which is the best starting hand”, or for that matter “how should you play each hand?” — we’ll get to that later. But this is just a math problem — which is most likely to win if you choose to stay in the pot all the way to the showdown?

The answer, to nobody’s suprise, is: it depends! It does not depend on your position, or whether the betting is limit or no-limit — those might affect your strategy along the way, but at the end of the hand it’s just a matter of who has the best cards. What it does depend on is how many people you are playing against. The absolute probability that you will win obviously goes down if you are playing against more opponents with randomly-chosen cards, just because there are more ways they could beat you. But, much more interestingly, the ordering of which hand is best also changes.

Here are the answers, presented in convenient tabular form. We’re showing the percentage chance that your hand will win outright, both against one other random hand and against four other random hands. The percentages come from running 500,000 simulated hands each, using the Poker Academy software. (It’s a very nice program, incorporating artificial-intelligence routines developed by the University of Alberta Poker Research Group. [Yes, there is such a thing.]) “Jd” stands for jack of diamonds, “Td” for ten of diamonds, etc. For later convenience we’ve chosen the ace to be the same suit as the JT, with all other cards being different suits (it doesn’t matter for this table, but does for the next one).

 Jd Td Ad 7c 6d 6h 1 opponent 56.2 57.3 62.8 4 opponents 27.3 20.7 17.9

So the miracle is that the relative strength of the three hands reverses when we go from one opponent to four. Against one other player, the sixes stand the best chance, followed by the A7, followed by the JTs (where “s” stands for “suited”). But against four, JTs is the most likely of the three to win, while the sixes are the least.

It’s not hard to figure out what’s going on. But before we do, let’s take a peek at something even more surprising. What happens if, instead of putting one of these three hands against some other random cards, we put them up against each other, two at a time? What is the relative ranking? Here is what happens:

 Jd Td Ad 7c 6d 6h Jd Td – 51.5 47.7 Ad 7c 48.3 – 56.7 6d 6h 51.6 43.0 –

The table shows the chance that the hand listed on top will beat the hand listed on the left side at a heads-up showdown (no other players). The entries don’t add up to 100% because there can be ties . So, another miracle: it’s not transitive! Sixes are likely to beat A7, and A7 is likely to beat JTs, but JTs is likely to beat a pair of sixes. It’s a kind of combinatorial rock-paper-scissors situation.

So what is going on? Note that if we consider just the two hole cards, without taking advantage of the community cards, the sixes are the best hand, followed by the A7, with JTs bringing up the rear. For one of the latter two to win, the community cards have to help it improve (by pairing one of the hole cards, or making a flush, or whatever). So the question becomes, how many ways are there to improve? The only likely way for the A7 to improve is for either an ace or a seven (or both, or several) to land on the board, although it’s also possible to find four board cards that help make a straight or flush. Adding up the probabilities, it’s almost a fifty percent chance, but not quite. Against the sixes, there are more ways for the JTs to improve. Both because the cards are “connectors,” allowing for cards that would give low straights (7-8-9) and high straights (Q-K-A) or various intermediate possibilities, and because the cards are suited, making it much easier to make a diamond flush. So JTs will usually beat a pair of sixes. But it won’t usually beat A7 if the ace is of the same suit. That’s because some of the ways that JTs will improve will also improve the A7 — in particular, if four diamonds come up, the JT will have a flush but the A7 will have a better one.

The same reasoning explains the first table. Against only one randomly-chosen pair of hole cards, there is a substantial chance that the sixes won’t need to improve, so they do the best; likewise the ace can often come out on top just by itself, so it’s second-best. But against four opponents, chances are excellent that someone will improve, and JTs has the best chance.

Which leads us to the other question: which is the best starting Hold’Em hand? It should be clear that there is no universally correct answer, and it will depend on game conditions — although, in ordinary circumstances, JTs is clearly the best, for a couple of reasons. One is that the thought experiment of playing your cards against another pair of randomly-chosen hole cards isn’t what really happens; in a real game you have a bunch of opponents, and the ones with weak hands simply fold, leaving only the stronger hands. So it’s almost as if you are playing against a larger number of opponents, even if a small number stay in for the showdown. The other reason (much more important) is that the criterion for success is not how many hands you win or lose, but how much money you win or lose. The A7 is not going to make you much money. If no ace comes up on the board, you’re likely beaten. If an ace does come up, either someone else has an ace with a better kicker (in which case you will lose a lot), or nobody has an ace and they will just fold (in which case you will win a little). Likewise for the sixes — if nobody can beat a pair of sixes, they’re not going to be putting much money into the pot. The only way to win big is if another six comes up, which is possible but unlikely, and you’d still have to worry that someone else made a straight or flush. This is why beginning players often over-value low pairs and aces with low kickers.

The moral of the story is that you don’t win in Hold’Em by knowing the percentage chance that your pocket cards can beat some other random two cards — you need to know what kind of hand your opponents are likely to have. Part of that is just probabilities, but much of it is gleaning clues from the way they have played the hand up to that point (did they raise, or call? how many bets? from what position?). In other words, you need a model of your opponents. Poker players have invented a simple two-dimensional parameter space of ways to play that serves as a simple model. One axis ranges from loose to tight — how often someone plays vs. folding. The other goes from passive to aggressive — how often someone simply checks or calls vs. raising. At the crudest level of analysis, you can locate an entire table of players at some point of the tight/loose and passive/aggressive plane; with a bit more data, you can describe individual players this way, and at a very sophisticated level you can get as specific as you like in an extremely high-dimensional parameter space (“they like to raise 80% of the time with pocket nines or better in fifth position with one bet and one caller before them when their stack is less than half of its starting value,” stuff like that).

That’s why it’s much harder to program a computer to be a championship-level Hold’Em player than a championship-level chess player. There is no perfect strategy in Hold’Em — no decision tree you could unambiguously follow to guarantee the best possible outcome. (Indeed, if you had an opponent that used such a decision tree, you could in principle always beat them.) Unlike in chess, the computer can’t win by brute force; it needs to be clever enough to learn from the previous moves of its opponents to figure out how they are playing. Teaching computers to play poker is an active area of research in artificial intelligence. And teaching humans is an active area of research in Vegas (although the “tuition” can get a little steep).

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