Disclaimer: I am not a mathematician. In fact, I have always been somewhat math impaired. This is especially frustrating to me because 1) I find mathematics quite interesting and b) you need at least a basic facility with math to be a good poker-player. And a broader understanding of certain kinds of math, especially probability and statistics, is especially useful if you want to really understand the kind of gambling you are letting yourself in for if you play poker with — how shall I say — intent
So, follow along with me here, and if you find any mistakes in my calculations, feel free to complain to my editor, because I am counting on her to do the fact checking. This is not my forté.
Okay, the first thing we have to talk about is probability. Probability is a concept that is so widely misunderstood and so easy to abuse in casual usage, that it’s a wonder anyone has a clue about it at all. For example: When the weather forecaster says, “Tomorrow, there’s a 20% chance of scattered showers in the greater metropolitan area" it means…
- One fifth of the greater metro region, by area, will have scattered showers.
- The last five times the weather chart looked like this, some areas received showers one of those times.
- I doubt very much it will rain tomorrow, but if it does, at least I have covered my ass by allowing for the possibility of a few showers.
- We’ve been keeping weather data for a long time, and doing our gosh-darnedest to draw out some correlations between conditions, trends, and the actual weather in the days following those conditions and trends. We have a lot of data, but we’re not sure how reliable it is. We do a lot of regression analysis. And we try to pretend that we have some idea how things are likely to go. The fact is, we’ve gotten a lot better at weather prediction, really we have. But for all practical purposes, “a 20% chance of scattered showers” means little more than “it might rain where you are, but probably not.”
Yeah, 5 points for answer D, 4 for C, and 2 each for A and B if the weather report preparer is not a meteorologist.
So now let’s talk about poker for a minute. Everyone wants to look down at the cards in their hand and see a pair of aces. Why? Because aces are 85% to win heads-up against any random hand, and 80% to win over any other pair. Pocket aces: yay! The best! You are gonna get PAID!!!! Right?
Well. Well, maybe
. Or maybe not
Ask any poker-player how they feel about “pocket rockets.” They love them. They want to get dealt them every hand. But they will also tell you sad story after sad story about how their aces got cracked. About the bad beats they’ve taken where some donkey with six-deuce offsuit drew runner runner for the straight, or how they were all-in pre-flop and the jack-four of hearts made a flush on the flop.
Pocket aces are the best hand in Texas Hold’em. And, over the long run, they lose 15% of the time. That’s a little more than one in seven occasions, on average. And believe me, when you’re playing for a monster pot or your tournament life, it seems like it happens a lot more often than that. Which is why, ridiculous as it may seem, there are players who actually say things like, “I hate the bullets, I always seem to lose with them.”
They don’t always lose with them, of course. But they remember the times they do lose, because it hurts so much, and they gloss over the times when they win with them, because they expect to win
with them. This is called “selective memory,” and it’s something poker-players should learn how to correct in themselves, because it has all sorts of pernicious effects. We’ll talk about that in other contexts too.
It is true, though, that your can get all your money in the middle pre-flop with pocket aces and lose four times in a row. Or three times out of five. Or eight times out of ten. You can lose with pocket aces over and over and over again, to the point were someone will quote “1:6.6” at you and you will laugh long and bitterly. When you look down at pocket aces you will see, instead, twin headstones with your name engraved on them, and you will long for the sweet, sweet release of death. You will develop a thirst for hemlock.
Welcome to the Land Ruled by the Law of Large Numbers, and welcome to its charming capital city: Variance (also known as Luckytown).
The problem with an 85% probability is that seven occasions mean nothing, statistically speaking. Anything can happen in a sample size of seven. Or 20. Or 200. Don’t quote me, but 85% doesn’t start meaning much of anything until you get into counts on the order of thousands, and even then the actual numbers might work out to describe a fuzzy bell-curve somewhere from 65% to 97%.
If you play 300 hands of poker in a typical large tournament, you’ll get pocket aces on average (here we go again with the probability numbers, but hang in with me) somewhere from one to three times. If you play three thousand tournaments
, you can expect that your aces will have held up just about 85% of the time when all is said and done. But in one tournament? All bets are off. You cannot say much more than “pocket aces are a very good hand to get all your money in pre-flop with.” Do it, and hope for the best. For heaven’s sake, don’t start keeping score and feeling entitled to have your hand hold up this time because you lost with aces five times in a row in the last few games. It just doesn’t work like that.
Probability is about numbers, but not just any old numbers. Probability is about really big numbers. Big, big numbers. Did I say big? Did I mention, huge? As in: very, very, large.
A program like PokerTracker, which helps players statistically analyze their play and that of others, starts to become useful with a data sample size of hand histories of at least ten thousand, and really comes into its own when you load up 100,000 hands or more. Before the advent of computer-based play, it could take a person well over a year, playing 8 hours a day, every day, to log that many hands (and of course they would have had no way — other than their memory — to keep track of all the action).
It is not unusual now, though, for active online players to log a million hands in a year. That’s a lot of data. Poker sites like Full Tilt and PokerStars have probably served up billions, by now. The statistics to be derived from a dataset like that should be very, very reliable, and should very closely approach the theoretical distributions that the math of the game predict. The correlation is so strong that any noticeable discrepancy between the two is likely to set off alarm bells about cheating or provoke cries of "rigged!"
There is, however, an acronym that crops up in most sensible comments in online forums that address statistics in single specific contexts: YMMV. "Your mileage may vary." It should probably read: YMWV. Because your mileage will
vary. Probability is deep. The probability of probability distributions is also probabilistic.
Let’s say we have a group of ten thousand monkeys, each of whom has been dealt and played 100,000 hands of automatic all-in heads-up poker. In that group of 10,000, each monkey’s statistics will vary from the expected theoretical distribution of outcomes by some amount. We would expect that distribution to look like a normal bell curve, with most of the monkeys clustered at the middle, and progressively fewer and fewer monkeys with outlying results either significantly more or less favorable than average.
We call the monkeys at the low end of the curve “unlucky” and the ones at the high end “lucky,” because they are “running good.” They are enjoying the upside of variance, which is the statistical deviation from expected probabilistic outcomes over a given sample size.
Every poker player wants to move from the left side of this graph to the right. Everyone wants to be the lucky monkey. Nobody wants to be the poor chimp at the bottom of this particular barrel.
Here’s the thing, though. For any given snapshot of population and timeframe, there is always an unlucky monkey. By definition, someone is going to get the short end of the stick. There’s just not enough positive and typical variance to go around.
Now let’s zoom out yet another time. Over a poker-player’s career, of — let’s say — a gazillion hands, one would expect standard probability to hold sway. And, ON AVERAGE, it will. But, guess what? There will always be the people who get the statistical shaft: somewhere out there, there’s undoubtedly a guy who always runs bad
. (With any luck, no pun intended, he’s already quit playing poker.) Similarly, there are at least a couple of people who always run good
Gosh, I wonder who they are?
Do you suppose some of them are the people who make it to the final table of major tournaments, or who have built enormous bankrolls online in remarkably short order?
That’s all skill. Skill
, I tell you! Poker is a game of skill