Wednesday, January 10, 2007

Wanting to Imitate Waffles, Puzzle Entries, Binomial Distributions and Other Matters

Enjoy a grab-bag post, all and sundry, just because I've been busy for a few days on several other projects. I did sneak in a trip to last night's WWnH (Wil Wheaton not Here) event last night on Stars, where I was disgusted enough at my exit to think about pulling a "Waffles." No definition necessary, of course. My departure wasn't a terrible beat, but when one puts a great read on someone, comes over the top of an aggressive player making a play and induces them into making a call against the odds, you'd like to see the poker gods reward the correct play.

If I win the hand, I'm the chip leader with 17 to go. I got the money in ahead, and it didn't work out. That's poker.

However, while congrats to the player who got me, whoever it was, I wasn't all that surprised to see that that player managed to turn a massive chip lead with 16 players left into an out-of-the-money finish. One is just not going to take the wrong end of the odds repeatedly against a CJ or a Hoy or a Smokkee or a Lucko and expect to come out on top. Or at least one shouldn't expect such a result. And that's the secret to why these players do so well; over time, they just get their chips into the pot with the best of it when it matters the most.

In a way, though, the suckout wasn't much of a surprise. I've been running very, very, very, very bad on Stars lately. It doesn't matter what game I try, because (at least on Stars), I'm in one of those dog-ugly variance downturns. WillWonka had a front-row seat at someone laying four consecutive suckouts on me at a triple-draw table a few days back, which is just the way it goes when one is in a rut. I've seen the other side of the equation, too, many, many times.

It's a good segue, of course, to a poker segment where I have been running well, those Bodog $15+1 turbo SNGs. In my previous post, I mentioned how I'd played some 68 of these and cashed at a 43.8% clip, and I wondered how the trend would hold as I played a few more. I also mentioned something called "signature significance," which in this context deals with small sample sizes. How can one look at 50 or 100 iterations of anything and determine if results are luck or skill?

It's a vital question for the poker player looking to analyze results. I was talking on the phone with one of my bestest poker friends two days ago, and we discussed some of this arcane mathematical stuff. Now, virtually every poker player inherently understands the concept behind the Law of Large Numbers: we know that 500,000 poker hands is a much truer test of poker ability than any 5,000-hand sample could ever be.

But there are still ways to look at smaller sample pools, especially in instances where one can tabulate a distinct departure from expected norms. The Bodog SNGs are 10-player events, so the chance of finishing in the money, all things being equal, is an easy-to-compute three out of ten, or 30%. We can use this knowledge to calculate something called a binomial distribution, which produces a bell curve of possible results based on the frequency with which they occur. There are lots of automated binomial-distribution calculators on the Web, too.

I actually had 106 of these turbo SNGs in the bank when I ran this calc, and had cashed in 46, for a tiny dip to 43.4% [46/106] ITM rate. Due to several firsts, though, the ROI actually snuck up a couple of percent over the greater sample.

To run the calculation, one needs to know three things:

N = Number of iterations (sample size)
P = Probability of a successful result
X = Number of times a successful result has occurred

We know all three of these things from the sample. N = 106, the number of these turbos that I've played; P, for the purposes of this calculation, is set to 0.3, because 0.3 (or 30%), is the exact expected long-term success rate for the average player within the context of this game; and X = 46, the number of successes (in-the-money finishes), achieved within the scope of the sample.

The calculator I used has more complex capabilities, but for this simple version of a poker-success stress test we'll just use the "Exactly" option, which will show us exactly the odds that a player with an expected success rate of 30% would actually achieve that success on 46 of 106 tries.

Now, Rocky, watch me pull a rabbit out of my hat!

The odds of a 46-of-106 performance for a player expected to perform at the 30% success level is .0011, or a bit greater than 900:1. On the graph, it's that little red tick mark well to the right of the big part of the bell, and also oitside the standard-distribution markers, those bolder vertical bars. 908:1, to be exact, and them's long odds. While there is a much greater likelihood that I'm a 37% or 38% player enjoying an extended rush, there's not much more than one chance in a thousand that I'm actually an average player compared to the other players at these stakes, considering the exact combination of circumstances and factors that define these turbo SNGs. The conclusion: Whatever my strategies are for these things, they work.

Nothing is ever for certain in a statistical sense, but even small sample sizes can cough up meaningful information. This is just an example of how a statistical evaluation can be used to interpret smaller samples of poker results.

And yes, as I've said before, it's comforting at a time when the Stars multitudes are taking turns kicking me into submission in other games. Also, yes, the Bodog plus is significantly more than the Stars minus, despite my whining.

Wrapping things up. Total solved crossword entries received to date: 8. Thank you to all who have sent in entries, and the contest remains open through the 15th.


Anonymous said...

The calculator I used has more complex capabilities, but for this simple version of a poker-success stress test we'll just use the "Exactly" option ...

Putting on my ex-Stats 100 lecturer hat, it would probably be more sensible (and certainly more standard) to use the "At least" option. As you can see "Exactly" even applied to 30 wins wouldn't give a very big number.

But the results are significant in any case -- well done! I do get annoyed with a fairly big chunk of the poker playing population (and I'm thinking in particular of posters on a well known forum) who tend to insist on ridiculously large sample sizes. They have a point with respect to hand by hand results as the variability there is so huge, but for these simple binomial outcomes, 100 trials, or a few 100 trials is generally more than adequate.

Haley said...

Good points, Michael, and good blog you've got, too.