Pandering Frivolity

By Sean Carroll | July 6, 2006 10:56 am

The baleful eye of the establishment media has once again turned our way, and judged us to be sordid muckrakers. Declan Butler at Nature has written about the largest science blogs, and we were happy to find CV in the top five, along with Pharyngula, The Panda’s Thumb, Real Climate, and The Scientific Activist. (Plenty of room to complain about methodologies, but whatever — suffice it to say that prize money was distributed quite equally.) The Technology Chronicles, however, has poked a stick at these would-be science blogs, and found that they succeed not by “politely debating the fine points of string theory” (ahem), but rather by “channel[ing] the static and political undercurrents in their fields.”

Nonsense! We have succeeded by writing about martinis and the World Cup. To cement our reputations as light-hearted bons vivants, today’s post is about poker.

In particular, a quiz. For those of you not addicted to Bravo’s Celebrity Poker Showdown, the game that has swept the public’s consciousness is Texas Hold’Em. It’s just a particular variety of poker, in which each player gets two hole cards that only they see, and then five cards are dealt face-up in the middle of the table. The winner is the one who can construct the best five-card hand out of seven — their hole cards and the five on the board. Complications arise from the baroque betting structure (two players to the dealer’s left are forced to bet on the first round, which is after the two hole cards are dealt; further betting rounds after the first three board cards are dealt, another after the fourth, and a final one after the fifth), but basically it’s just that simple.

So, consider the following three possible pairs of hole cards:

  • Jack-10 suited (e.g., a Jack of diamonds and a 10 of diamonds)
  • Ace-7 unsuited (e.g., an Ace of spades and a 7 of clubs)
  • Pair of sixes

The quiz is extremely simple, and should be easy for experts: assuming you don’t know what anyone else has, or yet what the board cards will be, which possibility is most likely to win at the end of the hand? And (a subtly different question) which is the best Hold’Em hand? Please show your work; answers will be revealed tomorrow. Winners will receive a free lifetime subscription to Cosmic Variance, one of the most popular science blogs on all the internets.

  • Elliot


    How many players at the table?

  • Sean

    Ten is the standard number, but of course some might fold.

    No more clarifications; ambiguity is part of the real world.

  • Garrett

    The straight answer would suit best.

  • JustAnotherInfidel

    Limit or no limit? It makes a pretty big difference…

    This is a tough one–well chosen. Playing things like A7 unsuited gets a lot of people into a lot of trouble! Against a table of ten hands, I will fold A7 offsuit about 95% of the time. If I call, I’m gambling or hoping to bluff later.

    The smart thing to do is call with the J10s if a lot of other people have just called the big blind, unless you’re playing limit hold em, then you should probably raise against several callers to get more favorable odds. If there aren’t any callers then your hand has probably lost some of its value. Most pros would rank this hand first, I think. (But who am I?) The reason being that straights and flushes are very powerful hands in hold em, because most people will never throw away three of a kind. You do have to be careful, though, that the straight or flush you make with JTs isn’t beaten by a higher one, a definite possibility if alot of people see the flop.

    As for the 66, you’re really looking to see a cheap flop, and hit something like three of a kind of two pair, or three low cards like 2 5 7. This doesn’t happen very often, and your worries are compounded by the fact that in low limit poker people tend to slow-play big pairs, and silly hands like A7 unsuited… You can’t really call a raise when more than two cards higher than a six come on the flop. In no limit, if no one has called, you might make a big raise in late position hoping to take the pot down right there.

    If the three hands are heads up, I think the odds probably break down in favor of JTs (40%ish?). Best hold em hands: JTs, 66, A7. In that order.

    Dr. Carroll—perhaps you could throw in an autographed copy of your GR lecture notes?

  • Elliot

    Infidel got it right in rank order. But the beauty and subtlety of the game is that it is completely situational. Hence my question. Heads up the 66 is actually a slightly better hand. Chip count, table position, number of players, playing style all affect each and every hand despite the well known and understood mathematical odds.

    Poker Trivia Fact for the day – Doyle Brunson is nearly a scratch golfer. For those of you who don’t know who he is. He is probably the greatest living legend in poker. In his 70s and about 100 lbs overweight. (not your average golfer)

    So here’s a follow up trivia question. Which hand is called the Doyle Brunson? and why?

  • Constance Reader

    I’m answering as a hardcore competitive Hold’em player, not any sort of scientist.

    First, a correction. The correct term is “pocket”, not hole.

    Play the pair, always play a pocket pair at least to the flop. Then make your decision to continue or not, based on the betting. You should probably baby bump the bet on this one, to chase people out pre-flop.

    Play the jack-ten suited, but not for the straight or the flush. Play for the chance of pairing or tripping one or the other. You could baby bump this one too, and not just as a chaser. You really want to see that flop.

    Some would tell you to always play the ace, but I am not of that ilk. I’ll play the Ace with a high card always. I’ll play this one to the flop if I have a comfortable stack, and/or if nobody raises the bet. Unfortunately you don’t alway have this luxury, some tables will always have one of those jerks that will raise on any trash he has in his hand, thinking nobody else notices.

    And the Doyle Brunson is the 10-2, because he’s won the WSOP two years in a row with it. I’d never play the 10-2, but then, I’m no Brunson.

  • Sean

    Constance, you didn’t actually answer the first question: which hand is mostly likely to win a showdown?

  • JustAnotherInfidel

    Yeah it is situation dependant. So if there’s only two or three people at the table, the A7 becomes much more playable. Also if you’re playing in a tournament and you’re near the pay-off line (where people tend to get ultra conservative), you would aggresively play questionable hands, like A7, that you wouldn’t have ordinarily. I still stand behind my plays at a full table, in a casino game.

    PS…If you read Super/System, then you know the DB is also and AQ offsuit, because he folds it every time. That one never quite caught on.

    Since this came up in the context of celebrity poker, who is the better announcer—Phil Helmuth or Phil Gordon?

  • Elliot

    I imagine Constance, you would consider me “one of those jerks”. I tend to play a lot of “crappy” hands I shouldn’t based on the odds alone, particularly early when the blinds/antes are low and when in position, if I can get a relatively cheap look at the flop, even throwing in a raise for effect. 42% of the total information about the hand is in the flop(3/7). If I can get that information for the “right price” I’ll do it all day long.

    I guess that’s what makes it such an interesting game
    Different Strokes for Different folks.

  • Chad Orzel

    assuming you don’t know what anyone else has, or yet what the board cards will be, which possibility is most likely to win at the end of the hand?

    The one I’m not holding.

  • NL

    which hand is most likely to win

    Hm. Let’s assume it’s 3 players. Against each other. 10 players complicates things and is unrealistic.


    66 is most likely to *lose*, as the probability is ~75% [1-(38,5)/(50,5)] that the board will pair one of the 4 overcards, and the chance of trips, the only likely improvement for 66, is ~20% [1-(48,5)/(50,5)]. Conditional probs should be small perturbation, and a host of other possibilities only make things worse for 66.

    Given that, A7s will edge out JTs due to a) the strength of the overcard and b) the superior flush draw edging out the crippled straight draw. I’d guess that it’s no better than a few % edge, though.


    which is the best holdem hand

    It depends. Now it’s more intuition-based… Despite my argument above, I’d have a tough time playing A7o in any situation, since most hands (when they’re played to the end) are heads-up, in which case things change significantly. I would prefer to play with the J10s, as they’ll give you more opportunities to make money. In a tournament I would play the sixes all the time while the blinds were low, ignore them in the midgame, and use them as desperation/miracle plays in the endgame.

  • NL

    Aw, hell, it was A7 unsuited.

  • NL

    You know, in a 3-handed game, I stand by that judgment. The overcard is just too strong. A7o edges JTs heads-up.

    That said, my situation of playing the 3 hands against each other is too precious by half. The real question is, which hand is stronger against N unknown opponents, where 150% chance of improving to be of Quality or Dominant.

    A7o has little potential, and will only become of Quality

  • NL

    Stupid less-than sign. I have no desire to write that post again…

  • Sakura-chan

    A7 all the way, and I don’t even need the math to back it up. =D

  • Pingback: Weltmeisterschaft | Cosmic Variance()

  • Constance Reader

    Sorry, it is situational. Where am I sitting in relation to the dealer, what’s the bet when it gets to me, how many have bet, what is my chip position, what are the blinds (the higher the blinds, the more conservative my strategy), what has been the betting patterns of the other players? That’s the real challenge of Hold’em, is all the factors involved. Counting the cards and knowing the percentages does you some good at the final table, but never very much. I’ve tried the playing percentages strategy, and it never works. Yes, you win some pots. But you don’t win big ones because if you bet/raise rarely, then you get callers rarely. The others know you’re holding.

    One of the fellows I play in my league has a tell and I’ve figured it out, because I find it irritating. Of course, they’ve all figured out that the more tired I am, the better I play, because if I lose I can go home and sleep. And I can’t hide fatigue.

  • star fish

    If the hands are played “hot and cold” — i.e., no betting post-flop, and all three hands (and only these three hands) going to showdown — then the ordering is JTs (~42% equity), 66 (~30%), A7o (28%). Of course, these numbers are pretty irrelevant unless, say, those three hands happen to get all-in before the flop. This, in turn, could happen in a tournament, with a couple of these players short-stacked, but is otherwise unlikely.

    The question of which hand is “best” is quite different: in practice, you care not just about how often a hand will win if it makes it all the way to the showdown, but about how likely you are to actually make it to showdown, how much action you will get when you’re ahead (versus how much you’ll give before you know you’re beat), etc. In ring-game (that is, a full 9 or 10-person table) limit poker, for instance, I would generally rank 66 above JTs for most players, with A7o far behind both. I’ll briefly explain why. These comments are for limit; no-limit ring is subtly different, and tournament no-limit is another beast entirely.

    66 can be played simply, and profitably, as a “flop it or drop it” hand: limp in (that is, call a single bet) with it from any position, and hope you flop a set (i.e., hope another 6 appears on the flop). Played in this way, 66 is profitable because you don’t lose much when it doesn’t hit (one bet), but you stand to make quite a bit when it does (a flopped set will usually hold up to win the hand). In poker parlance, the hand has high “implied odds” — you’re probably not getting immediate pot odds to play pre-flop, because you’ll only flop a set or better about 1 time in 9 … but that time you *do* flop a set you’re probably going to make a good bit of money. (An expert limit player can also pick up some additional profit from figuring out the occasional times when 66 is the best hand unimproved — but that’s tougher.)

    JTs is a fickle hand, but certainly profitable if played reasonably well. It makes the most straights, and all those straights (using both the J and T) will be the nut straight. The opportunity to make nut straights is nice: again, you’re looking for opportunities for someone to pay you lots of money when you have a lock (or near-lock) on the hand. Suitedness also helps (JTo is a substantially worse hand), even though you’ll only flop a flush draw about one time in 9. On the other hand, you don’t have a ton of high-card power — if you flop a T or J and anyone else is betting, you will have to worry that someone has AT or AJ and you’re basically just spewing money at them. Similarly, if you do flop a flush draw, and then make a flush, you will have to at least entertain the possibility that someone has a better flush — you’ll be drawing to the 4th-nut flush. For these reasons, you can’t bet with a ton of confidence; put another way, you can lose some money before you know you’re beat, but may not make a ton when you’re ahead. (Think about how much action you are really going to get from good players on a KQAxx board.) All of this is a long way of saying that the hand is fine, but easily overrated. (I would limp with it in late position in just about any game, but only play it up front in a soft game — you want lots of limpers behind you.)

    A7o, in contrast, is a sucker hand. It looks great — ooh, Ace! shiny! — but it’s never going to make you a ton of money, and can easily end up costing you a bunch. The basic problem is that you will often make second-best hands with it. If an A flops, one of a couple things will probably happen: either you’ll get no action, because even people with decent hands are going to fear the overcard, or you’ll get action from another A (or some other made hand or good draw). If someone else has an A, you are probably dominated: good players will come into the pot with hands like AT-AK, not hands like A2-A6 (unless they are suited: more on this in a second), but you’re going to want to call bets all the way to the river, where you will probably lose. How are you going to bet this hand with confidence? A: you can’t. You’re actually probably better off if you flop top pair with the 7, not the A: at least then you can get action from overcards (that guy with JT, for instance). On the other hand, those 7s aren’t going to hold up all that often. Note that A7s is actually playable (I would play it even from EP in essentially any limit game): you won’t make the nut flush very often, but when you do it can make you a bunch of money (since that guy with JTs is going to pay you off in a big way if he also made a straight or flush). And so on.

    The bottom line is that the hot-and-cold numbers are very misleading: the fact is that you’re not going to make it to the river with all these hands an equal number of times, and you’re not going to have put the same amount of money into the pot in each case.

  • Pingback: Poker Quiz Answers | Cosmic Variance()

  • trippnballz

    1) J10 suited is the favorite

    2) pocket 6’s is 2nd best

    3) A7off is about 1% worse than the pockets


Discover's Newsletter

Sign up to get the latest science news delivered weekly right to your inbox!

Cosmic Variance

Random samplings from a universe of ideas.

About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


See More

Collapse bottom bar