In November, 2009 I purchased a poker software program called Poker Tracker 3 (version 3.06.2), which is a great tool for the serious player, but also comes at a serious price: $90 for the full version that covers Hold’em; less for micro-stakes players and more if you add Omaha to it. You can find it at www.pokertracker.com/ Let me add that I purchased my copy at full retail and have received nothing from them for writing this.

In fact, this is not a review of the software – I like it and feel it’s worth the investment – but what I want to do here in July, 2010 is analyze my results from playing 380 tournaments at Poker Stars – not so much the $$$ won or lost (I’m down $285, which is a -8.73% return on investment, roughly the cost of the entry fees, which average a little under 10%) – but more as an analysis of the starting hands I received with an idea of discussing two popular thoughts about online Poker:

A. Is it dealt in a fair manner?

B. Is Poker truly a game of skill?

I’m not necessarily going to provide you with a concrete answer to these questions; if you think the game is rigged, you’re probably going to still feel that way when done reading this. As for Poker being a game of skill, I think it’s easier to say, “Yes, it’s a game of skill,” at least insofar that a game like Blackjack is a game of skill so long as a player does not double-down on 20s or consistently hit his hands of 17 or more. In other words, any game of skill can be played in a destructive way that assures the player will not win, but I’m assuming the average Poker player is motivated by profit, which seems to mitigate any desire to self-destruct. But – and this is a big but – I have become convinced that “luck” (or variance as we prefer to call it) plays a much bigger part in Poker than it does in a game like Blackjack.

I used to think that skill was the primary factor in winning at Poker – maybe 75% skill and 25% luck – else why do we see Poker pros like Daniel Negreanu and Phil Helmuth, et. al. make it to so many final tables of tournaments? Luck doesn’t last forever, right? Well, right – but luck in Poker can last a very long time just because of the huge number of permutations of hands one will see in the course of 20 or 30 or even 300 tournaments. While this is a bit outside of the scope of this lesson (I’m working on a follow-up article entitled, “Am I Good or Just Lucky?” that will explore the idea throughly), bear with me as I present some basic numbers that compare Hold’em Poker to Blackjack.

Blackjack, as we know it, is actually a very simple game when compared on a statistical basis with Poker. In Blackjack, there are only 39 different starting hands: 13 pairs, 10 “soft” hands – those with A-x, including ‘blackjacks’ but not including A-A, which is in the pairs category – and 16 “hard” hands that range from 5 to 20 in their totals (there is no hard 21 that can be made with two cards) and that’s primarily because suits play no part in a typical Blackjack game. Of course how one plays their hand is largely determined by the dealer’s up card, which can be one of ten cards, namely 2, 3, 4, 5, 6, 7, 8, 9, 10 and Ace. (Remember that 10s, Jacks, Queens and Kings are all “Tens” in Blackjack), which leads us to 39 x 10 = 390 different permutations we must learn to play. Sure, we also need to learn when to double or split pairs and so forth, but hopefully you see my point here, which is that Blackjack is a game of nearly complete information; the only things we don’t know are the dealer’s (single) hole card and the order in which the remaining cards in the deck or shoe will be dealt. Tell me what the next card is and I don’t care what the dealer’s hole card is. Conversely, tell me what the dealer’s hole card is and I don’t care what the next card in the deck is. Either way, I’m going to play one helluva profitable game! But I digress.

Now let’s look at Hold’em Poker, where suits do matter and each player is also dealt two starting cards just like in Blackjack. From a deck of 52 cards, there are 1326 possible two-card starting hands, but those can be broken down to “only” 169 starting hands because A-K suited is the same whether it’s in clubs, spades, diamonds or hearts. But we do make the distinction between suited and unsuited when it comes to starting hands in Hold’em Poker and those suits matter very much after the flop, but I’m getting ahead of myself here. In Blackjack, we’re competing only against the dealer; what cards the other players’ have makes little difference to us (unless we’re counting the cards of course but even then it’s not really important beyond the count if we’re playing a multi-deck game). But in Poker – let’s say at a table with 6 other players – their cards can have a huge impact on our hand as it plays out because we’re competing against them directly. So although we’re up against only five other hands, those hands can be from a “universe” of 169 hands (yes, we have one of those 169 but our A-Ks may be just one of three other AKs dealt on that hand) so the math tells us that we will be up against five hands out of a possible – are you ready for this? – 1,082,239,158 hands!

The math here is that there are over 1 billion ways to select five two-card starting hands out of a possible 169 hands. Put it in Excel as =combin(169,2) if you want to see if for yourself. With 40 cards remaining, there are 9880 three-card flops possible, which will leave 39 possible turn cards and 38 possible river cards. I’ll let you multiply all that out but it’s roughly 10,000 times one billion, times 39, times 38. While the order in which the cards arrive makes no difference from a strictly mathematical point of view, we’re obviously talking some huge numbers here. But to bring things down to earth a bit, so to speak, let’s say you have the aforementioned A-K suited in hearts. If the flop comes 2h, 3h, 9h you’re in pretty good shape and will likely win the hand. But let’s say you check because you’re hoping someone will bet and try to “steal” the pot. And sure enough, the maniac in the Big Blind bets out. Everyone else folds and you just call. The turn card is the 6h, but you’re not really thrilled to see it because it’ll likely kill your action; with four hearts on the board your opponent has to be concerned that you have the Ace of hearts, yet he goes all-in!

“Wow”, you think. “He’s either trying to buy the pot or he’s nuts.” Or, I would add, your opponent has a lower flush or has the 4h, 5h for a straight flush. It doesn’t really matter what cards your opponent actually holds; play often enough and you’ll see everything. Now I know what you’re thinking: I’ll never play a billion hands of poker so I probably won’t really see “everything” and that’s exactly my point. Even if you were to play a billion hands of Poker (remember that a billion is one thousand million), the universe of total hands is so big in comparison that even though you can expect to receive a hand like AK – suited or unsuited – once every 83 hands on average (and will likely show a long-term profit on the play of that hand, whether you’re a good player or not), you might easily lose the next time you get it and what if that’s your final hand in a huge tournament where you were one of the chip leaders but now you’re out? Bad for you and good for your opponent who might now go on to win it. Sure, “That’s poker,” as we all say, “I’ll get ’em next time.” You just have to play more hands and your edge – if you really have an edge – will eventually kick in and you’ll win your fair share of the $$$. But what exactly is “more hands?” Is it 600 more, 1000 more, 100,000 more? Think about it while we do this:

Tournament Hands: An Overview

Let’s now turn to some specific numbers, namely the 32,204 hands I played in 380 tournaments at Poker Stars over the past six and a half months.The Poker Tracker software captures all of the data on every hand I play, like the hole cards I’m dealt and what I did with them, such as fold, raise, call and so forth. How I played the cards is a factor of my skill (or lack thereof) but which cards I received are a function of the random number generator in the Poker Stars client (their software that resides on my hard drive.) Remember that “random” means without a pattern, so I’m not expecting to get AK on hand number 83 and again on hand number 166; that would be possible, certainly but highly unlikely. More realistically, I’d expect to get AK 388 times in 32,204 hands, which is 32,204 divided by 83 and rounded appropriately. I actually received it 412 times and did indeed make a profit – at least in tournament chips – from that hand. But in the case of another pretty good hand, A-J, it’s the hand I’ve received fewest times; only 327. Both of these hands are outside the “expected value” of 388 but not so far away to be statistically significant – that’s just poker. What’s my most common hand? How about the fabulous Queen-4, which I got 427 times. Ugly.

Pocket Pairs

Now let’s take a look at how I did in pocket pairs. You’ll hopefully recall that we can expect to receive any pocket pair once every 17 hands (a probability of 0.059 or 5.9% of the time) and a specific pocket pair like A-A once every 221 hands, which is a probability of 0.0045 or less than one-half of one percent. Here are my actual pocket pairs received in 32,204 hands where the expectation is to get 146 of each:

As you can see, I’m slightly above expectation in receiving A-A, which is obviously a good thing, and below expectation in receiving 2-2, which is also a good thing as far as I’m concerned. In an overall sense, I’ve received 1813 pocket pairs but I expected 1894 (32,204 divided by 17), so I’m showing some variance in the neighborhood of 2 Standard Deviations, which says there’s only about a 5% chance of this happening in a fair game. Do I think the game is unfair? In a word, no. But I am watching this statistic very closely, along with another one that I call “Top 19%.”

The Top 19%

The Poker Tracker software allows me to “filter” my starting hands in an unlimited way. Beside pocket pairs, I also track the top 19% (actually 18.85%) of all hands that I receive. These are all pocket pairs from A-A to 2-2; A-Ks to A-3s; A-Ko to A-10o; K-Qs to K-8s; K-Qo to K-Jo; Q-Js to Q-8s; J-10s to J-9s; 10-9s to 10-8s and 9-8s.In 32,204 tournament hands, I expected 6070 “top 19%” hands but have received 5855 which is a bit disconcerting because it’s a shortfall that translates into a 3.07 Standard Deviation event. One SD covers what will happen about 66% of the time; two SDs cover 95% and three SDs cover what will happen 99.7% of the time and I’m actually outside that parameter. From a simple mathematics point of view, the probability of my results are about one in a thousand for a fair game. Two points to consider: # 1, this is a relatively small sample and # 2, somebody has to be the “one-in-a-thousand” player, so why not me? (Although I swear I did nothing to deserve that!)

But when you really look at it, these numbers come from 380 tournaments and I’m short 215 “top 19%” hands, less than one per tournament, so I’m thinking that Standard Deviation may not be the best measurement for this statistic. Honestly, I don’t know of a better one but if you do, please email me. In the meantime, I keep hoping for some “regression to the mean”, which is a fancy way of saying things might start trending toward normal before I get to, say, the 60,000 hand level. It already has improved somewhat, because my worst point was back in April when I had played 21,863 hands but received only 3883 of the top 19%. The SD measurement then was 4.12, which really freaked me out because that’s about a 1 in 10,000 event – maybe more because my “Z” table doesn’t go that high! But it soon recovered and I began heading toward where I am now. I track this all the time and I’m not about to give up on playing Poker, so we’ll just have to see where it goes.

Now bear in mind that my results are to the downside and you can be sure that somebody out there is having a similar experience to the upside; that is, he or she is getting top 19% hands far in excess of expectation and may not be aware that it’s happening.To this player, the Aces are coming at just the right time, the Straights are being made, the Full Houses are filling on the river and so forth. If this is a new player, s/he believes that’s how poker goes – it’s not such a tough game, after all. And that thought is what has me working on my big article about luck. I’m reminded of a line from that great book, “The Black Swan”, which you should definitely read: “Lucky fools do not bear the slightest suspicion that they may be lucky fools.”

Other Interesting Observations

But let’s move on with an analysis of the 32,204 tournament hands that I’ve played thus far. My most profitable hand is A-K. Yes, it beats out A-A primarily because I’ve received, as mentioned before, A-K 412 times and A-A only 147 times. On a “Big Blind per hand” basis, a figure that the software calculates automatically, A-A wins 1.52 BB per hand while A-Ks wins 1.13 and A-Ko, surprisingly, wins 1.44.

Of the 169 possible starting hands, only 45 show a profit; the other 124 are losers to one degree or another. The biggest losing hand – in terms of tournament chips lost – is A-4 offsuit.A shock to me is that the second-most losing hand is Q-J suited! And it’s been that way for quite some time. But on a Big Blind per hand basis, it’s a winner to the tune of 0.84 BB. I must have lost a whopper of a hand with Q-Js in a multi-table tournament, which could explain the net loss of chips while building up a nice BB per hand statatistic on a hand that ought to do pretty well overall.

To wrap up here, I really do think the game at Poker Stars is honest but while I trust, I’m also going to verify. And I am more and more convinced that “luck” or variance plays a much bigger part in the game than we realize. I hope to quantify that in a future article – like I mentioned at the beginning – but my guess (an educated guess to be sure, but still a guess) is that skill makes up less than 20% of the game.

I’ll see you here next time.