Showing posts with label poker stove. Show all posts
Showing posts with label poker stove. Show all posts

Tuesday, October 09, 2007

The worst hand in poker--are you sure?

A couple of weeks ago, while playing at the Hilton, I was in the big blind with 2-3 offsuit. The pot wasn't raised, so I got into the hand for no extra money. I got no help from the community cards, and at the end was playing the board, because all five cards out there were higher than mine. I got to the showdown because the action went check-check-check--it was one of those pots in which nobody made any hand worth betting on. I laughed as I exposed my cards, and said, "I don't think I win, since I have the worst possible hand."*

A player to my left said, "No, 2-7 is the worst possible hand."

I will cut him a bit of slack because he was referring to the worst starting hand, while I was referring to the worst possible hand at the end, given this particular set of five community cards.

But even after accounting for that misunderstanding of the context, the guy was wrong. Or at least he wasn't clearly right.

I first heard 2-7 (offsuit--and throughout this post I am granting that the hole cards are of different suits, because when they are of the same suit they automatically gain in potential value, due to the increased chance of hitting a flush) described as the worst starting hand in hold'em poker from Mike Sexton's commentary on one of the early episodes of the World Poker Tour, when I was just starting to understand how poker is played. And the observation stands to reason, because they are the lowest cards you can hold that can't make a straight. (That is, you can't make a straight using both cards--which means that in order to hit a straight you need help from four community cards, a considerably harder task than making a straight with just three of them.)

But anybody who makes the blanket claim that 2-7 is the worst possible starting hand just hasn't thought the matter through. In fact, next time I hear somebody say that, I think I'll test the strength of their belief by offering them a bet: "OK, I'll take 2-7 and give you 2-3, and we'll play out the hands, without any rounds of betting, just all five community cards at once. We'll do that, say, 100 times, and whoever has won the most hands gets $1000 from the other."

I sure hope somebody takes me up on that. Would you, dear reader? I hope not--because I hope my readers are smarter than that!

According to Card Player magazine's online poker odds calculator (http://www.cardplayer.com/poker_odds/texas_holdem), 2-7 beats 2-3 56% of the time, the 2-3 wins 26%, and they split the pot in 18% of hands. (All percentages in this post are rounded.) I like my chances for winning that $1000, if I can get anybody to agree to the bet.

As with nearly everything in poker, the answer to the question "What's the worst starting hand?" is "it all depends." Primarily, it depends on what you're up against.

Continuing with the 2-7 versus 2-3 comparison, let's see how they stack up against some sample opponents. I'm using two different tools for doing the calculation: the Card Player calculator and the application called Poker Stove (available for free download from http://www.pokerstove.com/). They come up with slightly different results, mostly because the latter doesn't spit out a separate percentage for ties, but takes the frequency of ties (e.g., the 18% above) and splits it between the hands being compared (in that case, it gives an extra 9% to each, and reports the result as 65% for the 2-7 and 35% for the 2-3). There are also probably differences in how the calculation is run, though I don't know the details of either one's operation.

In the table below, I show the results of pitting the 2-3 and the 2-7 against a variety of other hands, as shown. The results are, in each case, using the CP calculator first and Poker Stove second. The dollar sign shows which is superior (i.e., between the 2-3 and 2-7):

A-A versus 2-3: 86/13, 86/14 $
A-A versus 2-7: 87/12, 87/13

A-K versus 2-3: 65/34, 65/35 $
A-K versus 2-7: 67/33, 67/33

9-9 versus 2-3: 84/15, 85/15 $
9-9 versus 2-7: 88/12, 88/12

5-5 versus 2-3: 86/12, 87/13
5-5 versus 2-7: 70/28, 71/29 $

2-2 versus 2-3: 64/30, 67/33 $
2-2 versus 2-7: 65/30, 68/32

A-2 versus 2-3: 74/25, 74/26 $
A-2 versus 2-7: 74/25, 75/25

A-4 versus 2-3: 67/33, 67/33
A-4 versus 2-7: 65/35, 65/35 $

Which is kind of a long way of saying that against at least some other opponents' starting hands, 2-7 would be a better choice than 2-3, sometimes a far better choice.

(As an aside, neither the 2-3 nor the 2-7 is the worst hand to take up against somebody else's pocket aces. Top--or, rather, bottom--honors there go to the A-6 offsuit, which is a 94/6 dog in that race, which is the absolute worst scenario of any two hold'em starting hands. The point being, again, that what qualifies as the worst hand depends entirely on what one is up against. Or, to avoid ending that sentence in a preposition, it depends entirely on up against which one is. (Yuck. So much for grammar.))

Of course, we don't get to choose either our own hole cards or those of our opponents. So probably the best way of answering the question of the worst starting hand is by running the test against a random hand. This is a calculation that Poker Stove does easily, but isn't available from the Card Player tool. The answer is that the 2-7 beats or ties a random hand 35% of the time, the 2-3 only 33%. Matthew Hilger's book, Texas Hold'em Odds and Probabilities, gives these numbers, respectively, as 35% and 32% (pp. 235-237), a difference I assume is probably attributable to just rounding.

By that most general metric, then, 2-7 is a better place to start than 2-3. In other words, if what we're interested in is the likelihood of ending up with the best hand after all five community cards have been dealt out, the worst cards to have in your hand are what I had in the story that prompted this rant: a 2 and a 3, not the infamous 2-7.

So why does Mike Sexton say that the 2-7 is the worst starting hand? And since he's not exactly the first one to make that observation, why is it found in so many sources?

The answer is that nobody plays poker the way that the calculators do the math, or the way that I would propose to run my 100-hand bet.

Given the choice, I'd rather take a 2-3 than 2-7 into battle in any given pot, even though I know that 2-7 beats 2-3 a lot more often than it loses to it, and even though 2-7 will win slightly more showdowns than will 2-3 against random opposing hands. The reason is that I want to be able to make a strong hand against an opponent who has a hand that isn't quite as good but is good enough that he's willing to bet on it (and lose).

Considering the 2-3 and 2-7, they are equally likely to make one pair, two pairs, three of a kind, a flush, a full house, or four of a kind. But, as mentioned early in this post, they are not equally likely to make a straight. And that makes enough of a difference (because straights are relatively common) that I'd prefer to start with the 2-3. It is more likely that I will hit a straight. Of course, with the 2-7 any flush or full house I hit will be higher than making the same category of hand with the 2-3. But the increased probability of the straight easily outweighs that factor, in my view.

The probability of flopping a straight with the 2-3 is only 0.7%, according to Hilger (p. 189). But the probability of flopping either a straight or an open-ended straight draw is much better: 3.8% (Hilger, p. 194). With 2-7, it's obviously impossible to flop a straight. Hilger doesn't list the probability of flopping an open-ended straight draw with the 2-7, so I had to work it out myself: 1.6%.**

To make matters worse for the 2-7, having four parts of the straight on the board (as is necessary to make a straight with 2-7 in one's hand) is much more suspicious to an opponent holding, say, trips or two pair than is having just three parts of the straight visible. This translates to the 2-7 being less likely to win a big pot even when it does manage to make a straight. Furthermore, when you're relying on four parts of the straight being on the board, it's a lot more likely that an opponent will have a higher straight (or at least split the pot with you by holding the fifth card in common) than it is when that opponent has to have both of his hole cards exactly right to have a higher straight than yours.

In that sense, yes, the 2-7 is the worst starting hand, because it's the one I would least like to have to work with. I assume that most professionals have reached the same conclusion--hence the common observation about the poor, much-maligned 2-7. But now my readers can be among the cognoscenti, those who know that the answer isn't really that simple.

Now if I can just find that guy from the Hilton again, and offer him my little contest....



*Of course, any two unpaired cards in the hole that were both smaller than whatever the lowest card on the board was (I don't remember what that might have been) would actually be equally bad, since all such hands would split the pot, all playing the board. But I obviously had the lowest possible cards that fit that description.

** In case anybody wants to check my work: There are 19,600 possible flops (we don't care what order the cards come in), from the remaining 50 unknown cards (because C(50,3)=19,600). The flop has to be 3-4-5, 4-5-6, 5-6-8, 6-8-9, or 8-9-10 to give us the draw. Each of those has 64 different ways of hitting (because there are four cards of each rank), giving a total of 320 different flops that qualify. So the probability of hitting the draw is 320/19,600, or 1.6%.


Addendum, October 9, 2007:

After writing and posting the above, curiosity drove me to do a Google search on the phrase "worst hand in poker" to see what others have written on the subject. Along the way, I came across a table of percentage wins and pot equity of every hold'em starting hand, as played out (or so the author claims) in a computer simulation against a random hand in 2,000,000+ trials each: http://www.gocee.com/poker/he_ev_hand.html/