Practical Probability - Part V

There are certain statistical operations that you perform quite frequently in your daily life without really thinking of them in a statistical sense. Sampling Theory is a relatively complex division of statistics, yet it is something we routinely do. For example, a new restaurant opens up near your apartment. You drop in for lunch, and find that it is noisy and has slow service and mediocre food. You don't go back. Your sample size was one. Is that a big enough sample to be sure? Possibly it is. Let's now say your girlfriend says that she loves that restaurant, and the two of you go there for dinner on Valentine's Day. The food is great and the service has improved. Your sample size is now up to two. And without realizing it, you have preformed a stratified sample. Once, you went there for lunch, and once, for dinner. You didn't realize you were that smart, did you?



Basically, sampling is a technique designed to allow the sampler to draw conclusions about a "population" by testing a small group chosen from that population as a representative sample. There are two major types of errors that can be made in sampling. The first is choosing a sample that is too small. The second is choosing a sample that is not representative of the population. Looking at our simplistic example, there is no real way to tell what is the appropriate sample size for dining in a restaurant before you arrive at a rating that is probably accurate. Perhaps two or three bad meals would be sufficient. But what if you have two good ones and one bad? An even more difficult problem is deciding if your sample meal is representative of the population of all meals. We have already seen that lunch and dinner may be different groups within the population. Within each group, there may be subgroups – meat, fish, and pasta, for example. (It is very difficult for one individual to perform this type of sampling accurately, and that makes guides like Zagat extremely popular, since their sample is based on a large number of diners, eating different things at different times.)



When playing poker, every hand you play or observe becomes part of your sample. The more observant you are and the more often you play against someone, the more accurate the inferences you can draw about him and various situations. In general, when something happens frequently, a relatively small sample will be meaningful. When something is rare, you will need a very large sample to reach any meaningful conclusion. You will end up having the largest sample on yourself, and even this may be much too small to prove anything. Let's say you want to know if you have a big edge in $10,000 buy-in live tournaments. You decide to play 100 of these tournaments. So, your sample size is 100. These tournaments tend to have top-heavy payout structures and around 500 players. You believe that 100 is certainly a large enough sample to prove that you have an edge. Is it? Let's look at four players and see what percentage of the time they will win a tournament. One is a very weak player, and is a 1,000-to-1 underdog. The second is a break-even player, and is a 500-to-1 underdog (remember that we are assuming 500 players in each tournament). The third is very good, and is only a 250-to-1 underdog. The final player is a superstar, and is only 100-to-1. The following table approximates the percentage of the time that these four players will win anywhere from no tournaments to three or more.

0 1 2 3 or more

Weak Fish 90% 10% 0 0

Break Even 82% 16% 2% 0

Solid Winner 67% 27% 6% 0

Superstar 37% 47% 15% 1%



Winning no tournaments, batting 0 for 100, proves nothing. You are more likely to be a weak fish than a superstar, but even the superstar goes 0 for 100 more than one-third of the time. Winning one tournament also proves nothing. Now you are more likely to be a superstar than a weak fish, but nothing really significant has happened. If you win two tournaments, you can feel very confident that you have an edge, possibly a big edge. If you win three or more tournaments, you are almost certainly a very strong player, somewhere in the superstar zone. If the number of tournaments was increased from 100 to 1,000, the inferences we could draw from the results would be more accurate. If we included second- and third-place finishes, we could also increase our accuracy. The key point is that the two most likely results for 100 tournaments, zero or one win, prove nothing. Since these were $10,000 tournaments, we invested $1 million. Unless you are a superstar, your most likely result was to lose somewhere between half a million and a few hundred thousand dollars.



Let's look at cash-game results. Here, a sample size of 100 eight-hour sessions is big enough to be reasonably sure that any inferences we draw are meaningful. Anyone who wins more than one big bet an hour in low-limit poker or between one-half and one big bet an hour in bigger-limit poker is a winner. The corresponding wins in no-limit are two buy-ins per night at small stakes and one buy-in per night at bigger stakes. There are some caveats. I have seen some players start their record-keeping after they have had a few big wins, and they include those wins. This is a biased sample. If you decide to track your results for 100 sessions, specify it in advance, and then play the sessions. Another problem that could arise is lumping together sessions in a variety of games or stakes. If you play 95 sessions of $1-$2 blinds no-limit hold'em and five sessions of $10-$20 blinds no-limit hold'em, you really have an effective sample size of five. If you play 33 sessions each of limit hold'em, no-limit hold'em, and deuce-to-seven triple draw, you should treat each game separately and not add the results together. It is true that adding these 99 sessions together might show that you are a clear winner, but you'd be better off examining each category separately and drawing inferences about your performance in each individual game.



In this column, I looked at samples and sample sizes relating to your own performance. Even though you can accumulate a lot of data about your own play, it may not be enough to enable you to draw meaningful conclusions. This is especially true of situations in which the quality being tested occurs infrequently. Here is one final example of this from a non-poker situation to conclude this column. If you wanted to know what percentage of men are over 5 feet 8 inches tall, a sample size of 100 would give you a pretty good idea. If you wanted to know what percentage of men are over 6 feet 8 inches tall, a sample size of 100 might lead you to conclude that no one is that tall. In my next column, I'll look at drawing conclusions about your opponents.



Steve "Zee" Zolotow, aka The Bald Eagle, is a successful games player. He currently devotes most of his time to poker. He can be found at many major tournaments and playing on Full Tilt, as one of its pros. When escaping from poker, he hangs out in his bars on Avenue A – Nice Guy Eddie's at Houston and Doc Holliday's at 9th Street – in New York City.