Friday, March 23, 2012

Matching or non-matching suits?

In the March 26, 2012, issue of Poker Player Newspaper, Mike Caro's column has some questions and answers about probability. As one example, he asks the reader to consider the situation in which pocket sixes are pitted against pocket aces. Do the sixes have a better chance of winning if they match the suits of the aces, or if they are of different suits?

Caro knows the reader will expect the answer to be "different suits," because then the each of the sixes has a small chance of making a flush to win, whereas if the suits match, then any flush made with a six is automatically beaten by the higher flush made by the ace of the same suit.

Caro says (page 21) that this intuition is wrong:
And, no, it isn't better to have different suits if you hold sixes. That seems like the logical assumption at first glance, but it's the other way around. If your suits are both different, that's slightly worse than if one suit is duplicated. And one duplicated suit isn't as good as two duplicated suits, from the perspective of [the person] with sixes.

Although it's true that different suits means the sixes will win some hands with flushes that would have been impossible if the suits were the same, there's a second factor. And the second factor overwhelms the first. When your sixes are of the same suit as the aces, the defensive power overwhelms the offensive power. You prevent the aces from making enough winning flushes that you can afford to sacrifice all potential winning flushes of your own.

So, 6c-6d does better against Ac-Ad than 6h-6s does.
I was surprised by this. It is the opposite of what I had always assumed to be the case.

In fact, I was so surprised by it that I decided to do a blog post about it in the form of a quiz, asking readers this: If you have to be stuck on the bad end of a roughly 80/20 chance of winning (which is basically what all overpair/underpair situations come down to), which sixes would you choose to maximize your win percentage?

I fired up PokerStove so that I would have the exact answers available. But when I ran it, I discovered something interesting: Caro is wrong, and the conventional wisdom is correct.

Specifically, when tested against Ac-Ad, I find:

Hand Win % Tie % Equity %

6c-6d 18.66 0.22 18.886

6h-6d 19.36 0.18 19.542

6s-6h 20.05 0.14 20.199

This was run under the "enumerate all" option. Running a Monte Carlo simulation produces very slightly different numbers, depending on how long you let it run, but the pattern is always the same as shown above: The sixes with suits matching the aces do worst, those with different suits do best, and those with one matching and one different are about halfway in between.

I don't suppose PokerStove is infallible, so I did the same comparisons using an online poker odds calculator, this one on the web site. It generated numbers identical to those shown above. I also tried it using the calculator at, and again got identical results. This gives me considerable confidence that the numbers above are indeed correct.

So now I'm stuck with the real headscratcher: Why is Caro insisting otherwise? I cannot figure that out. If he had just said in passing that the underpair with matching suits does a little better than those with different suits, I would assume that there had simply been an error inadvertently introduced in the writing/editing process. Lord knows I've made my share of slips such as that. But Caro spends a good portion of the column explaining why the matching suits is better for the smaller pair. It can't be just a glitch along the lines of saying "better" where he meant "worse," or "same suits" where he meant "different suits."

I sent Mr. Caro an email three days ago, quoting his article and the numbers I found when I ran the simulation myself, and asking him to explain why he says that the opposite is true. I have not heard back from him.

I am genuinely baffled by this mystery.


Paboo said...

Odd Indeed.

Same suites, your flushes always lose. One different suit, those flushes will win. Two different suites, more flushes will win.

Those winning flushes will always be in greater number compared to you making trips/straights with same suits and AA hitting flush.

lightning36 said...

Great post, Grump. I will be interested in hearing response if you ever hear back from him.

Memphis MOJO said...

Nothing gets by the eagle eye of the Grumpmeister.

Anonymous said...

Good catch grump! "defensive power" argument also makes no sense because you are already behind the pair of Aces. Interested to read what Caro's (if any) response is. Maybe a letter to the editor of Poker Player Newspaper (is there a link BTW?)

Rakewell said...

In case this wasn't clear, I wasn't trying to find errors or prove Caro wrong. Based on his history as having been one of the first to use computers to generate probability tables for poker and my longstanding respect for his writing, I just assumed he was right when I read the article. I was pleased to learn something new. I was surprised to discover that he was wrong, and it actually took me quite a while to gain sufficient confidence that I wasn't the one missing something.

Fred said...

I almost remember reading a couple years ago about some flaw in the calculators, but it was so miniscule it almost didn't matter. I really can't elaborate at all on that, though, I don't remember what flaw the writer was referring to... To think about why same suits could do better than off-suits, maybe it would have something to do with the fact that with an off 6, you won't always win if your flush hits, only if your 6 plays. Some pots you'll split if your flush hits, but it's too high. Aces win all their flushes, so, by having the same suit as the Aces, you're taking 2 cards away from the deck that give the flush to them. While offsuit 6s win most flushes, they split some... Plus the Aces have all their flush outs available...maybe that's where he's coming from...the defense is better than the offense...?

James Antill said...

Not read the article, but what he says is true _if_ you are already at the flop and have hit a six ... but I'd guess most people would assume that to be the case.

And if you aren't getting all in pre. ... then having the same suits means that one of your flushes will hit less often, so you'll fold more post flop ... which is better for you (maybe this is what he meant).

Anonymous said...

Could he mean that he wants the aces tit a flush? Of so would that give you more chances or a full house to have a monstor hand beat a great hand?

James Antill said...

Article is here:

...looks like he was just wrong. Also a thread on 2+2:

Anonymous said...

Any updates from Caro or theories as to why this might be true?

Rakewell said...