2) pocket 6’s is 2nd best

3) A7off is about 1% worse than the pockets

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.

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.

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

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

A7s
JTs
66s

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.

Interesting…

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.