Science Behind Poker

The game of Poker involves a lot of odds and probability collection at every step along the way, and even though there is an undeniable psychological aspect to the game, math is still the predominant factor in Poker. In this article, we will take a look at the overview of probability and how it works in connection with Poker. WiFi Wars - Comedy Tour Show. Get your game face on – it's poker night at Science Oxford. Brush off your pack of cards and outplay Lady Luck with an evening of poker probability and psychology followed by a friendly poker tournament. Handling tilt is a multi-layered process, but it begins with understanding two things: why the tilt is happening and accepting that poker is a game ruled by odds. Yes, poker is a skill game, but it's also a game governed by the turn of a card. The sooner you accept that the luck of the cards can turn against you, the better you will handle tilt. Is it better to be lucky or good? Should we trust our gut feelings or rely on probabilities and careful analysis when making important decisions? In this quick talk, professional poker player Liv Boeree shares three strategies she's learned from the game and how we can apply them to real life. I spent part of last week on vacation from science in Las Vegas, where I thankfully avoided financial ruin due to some fortunate combination of genes, math awareness and a wife that has no interest in gambling.

1. Science Behind Poker Rules
2. Science Behind Poker Bar
3. Science Behind Poker Game
4. Science Behind Poker Games

How do most people imagine gambling? Waiting for your lucky opportunity, the perfect card, roulette number etc. I am sorry to disappoint you, but poker pros don't play that way. Successful cash game players will never base their decision on some 'magic' feeling or anything like that. Japanese slot machine hopper. All decision will be based on poker math and psychology.

On the other side, amateur players will gamble a lot. They will have their favorite hand, usually some garbage like Q9 or 25s and they will always play it because it is their lucky hand. They will have favorite dealer, favorite seat position, lucky charms in form of card protectors or anything else. Some players are even superstitious about the clothes they wear or even place where they parked their car. One player at the WSOP main event few years ago thought that his chewing gum was lucky so he kept the same gum for few days during the event. Disgusting and ridiculous if you ask me.

The basics

So how does poker math work ? It is actually quite simple. All poker games are played with a single deck of cards. There are exactly 52 cards, 13 of each suite and for of each kind (4 deuces, 4 kings etc.). So how can we know the odds of making our hand? Lets start with a simple example. Lets say that we hold AK.

So we saw 2 cards out of 52. We will make a hand with A or K on the board. There are 6 cards in the deck that we need ( 3 K and 3 A). The odds of the next card coming A or K is 6/50*100(to get percentage)=12%. So we will make our hand on the next card in 12% cases, but on the flop there are 3 cards so we can just multiply with 3. Odds of flopping a pair is up to 36% ( we can not know is it exactly 36% or less because someone can have A or K in their hand and that reduces the number of outs).

So tell me, how many times have you seen people getting angry when they miss a flop with AK ? Many times right ? And now you also know that you will miss most of the time and hit only 1/3 cases? Great, so there is no reason to get upset because of that. It is just math.

Poker math – odds on the flop

Lets see what outcome we'll get with a little more advanced example. Lets say that you hold KQs. Board is 237 with 2 cards of your suit. How often will you make a flush ? We're gonna use the same logic as before. We saw 5 cards out of 52. There are 47 cards left. There are 13 cards of each suit and we have 2 and 2 are on the board so we have 9 outs. So what are the odds of making our flush on next card ? Yeah, you guessed it correctly 9/47*100= 19%. And on the river ? Now we saw one more card so 9/46*100=19.6%. Combined odds of turn and river are little less then 19%+19%= 38%. In this example it is 36%.

Rule of 2 and 4

There is also a very easy shortcut you can use in middle of the game. Rule of 2 and 4. So number of our outs is 9. We will multiply with 2 for turn odds and by 4 for turn and river odds. So we will make a flush on the turn 9*2=18% and on the river 9*4=36%. Like you see it is not 100% mathematically correct but it is good enough to help you make a right decision.

Now it is time to plug in our opponent's hand also. Lets make a similar example like the previous one. Board is T63 with two spades and we hold KQ of spades. We suspect our opponent to have AT. Lets see if poker math is on our side. We can make a wining hand with a flush but also with K or Q. There are 9 flush outs and also 3 K and 3 Q. We have 15 outs so by using the rule of 4 we can see that our odds are roughly 15*4=60%. To be precise: in this example our odds are 55% but rule of 2 and 4 is a good enough guideline to help us realize that we are actually in lead even though we are holding only K high.

Common poker math situations

AAvsKK all in preflop =82%( same odds versus any smaller pair)

AKvs77=49% (same odds with any 2 overcards to the pair)

AKvsKK= 30%(same odds is AQvsQQ, AJvsJJ etc.)

A3vsKK =28%(same odds with one over to any pair)

Flop a set with pocket pair = 12%

Make a set on the turn = 4%

River a straight with inside straight draw (gut shot) =17%

Hit the flush with flush draw = 36%

River a straight with open ended straight draw = 32%

Make a hand with open ended straight and flush draw = 55%

River a hand with flush draw and inside straight draw = 45%

AKvspair on the flop = 24% (same odds with any 2 overcards to the pair)

Final thoughts

So would you bet money on your basketball game against Michael Jordan ? Probably not if you are not basketball pro. Would amateur poker players bet money against pro players in a poker game ? YES THEY WOULD. And they are doing it all the time. Why ? Because they can get lucky that one time or that one night. You can not get lucky in a basketball game that much to actually win the game. And what it means to get lucky in a poker game ? To play against the odds and win. So if you are betting on a 75% situation you are not guaranteed to win. You will win 3 out of 4 times in a long run but this evening you can lose 3 times in a row!

Always remember poker is not a sprint it is a marathon. Work on your math basics and your game and do not try to get lucky. Leave that to gamblers. Just learn to be a better player. If you want to continue learning try watching my instructional video -> NL5 zoom on pokerstars.

Good luck in your progress and may the odds be ever in your favor 😉

Last week, a federal judge in Brooklyn overturned the indictment of a Staten Island man who ran poker games in the back room of a warehouse, on the grounds that poker is a game of skill, not chance -- and hence, such games cannot be prosecuted under federal laws prohibiting illegal gambling businesses.

It's just the latest sally in the ongoing debate over poker that's been raging for more than 150 years. And it comes on the heels of a ruling last year by the Justice Department that 1962's Wire Act applied only to sports betting, not poker. This is kind of ironic, since the Justice Department also shut down online poker in the spring of 2011, charging the men behind the three most popular online sites with fraud and money laundering.

Clearly, the issue is far from resolved, but John Pappas, executive director of the Poker Player's Alliance, is encouraged by the latest ruling by Judge Jack B. Weinstein. 'Today's federal court ruling is a major victory for the game of poker and the millions of Americans who enjoy playing it,' he said in a statement. (The alliance is dedicated to decriminalizing poker.)

But wait! Via @Chemjobber on Twitter, I learned of a spanking new study by German researchers concluding that winning at poker is basically all about luck. They recruited 300 poker players, half self-defined 'experts' and half 'average,' sat them down at tables of six, evenly divided between expert and average players, and then had them all play 60 hands of Texas Hold 'Em. Oh, and they fixed the deals, the better to measure the effects of luck.

Their conclusion, per Neuroskeptic: 'Luck, rather than skill, was key in determining final balance, with experts taking no more, on average, than novices. Experts did play differently, on various measures, and seemed better able to cope with bad luck, losing less; but they also won less when given good cards.'

So are the German researchers correct that poker should thus be classified as gambling? Not necessarily. A 2008 study concluded that poker is a skill -- students who received some basic pointers performed better while playing 1000 hands of poker than those who received no training at all. Still other studies support the German conclusion. Who are we to believe?

Neuroskeptic rightly points out a major flaw in the 2012 study, namely, the classification of 'expert' players was based on self-reports. I would argue further that playing a mere 60 or 1000 hands of poker is an insufficient sample size, given the statistical complexities of the game. There are 52 cards, with more than 2.5 million possible five-card combinations. Texas Hold 'Em uses seven cards so there are around 133 million combinations. Plus, you know, fixing the deals really messes with those probabilities.

Compare this to the sample size of the expert witness cited by Judge Weinstein in his massive 120-page ruling. Randal D. Heeb is an economist and statistician (and avid poker player) who analyzed 415 million hands online of no-limit Texas Hold 'Em and found that the skill of a player 'had a statistically significant effect on the amount of money won or lost.'

The many mathematicians and physicists who are aficionados of poker would agree with Heeb. I wrote a feature for Discoverin November 2010 on poker-playing physicists, which included the Time Lord (a.k.a. Caltech physicist Sean Carroll, a.k.a. my Better Half), as well as string theorist Jeff Harvey, particle physicists Michael Binger and Marcel Vonk (both of whom have done extremely well on the professional circuit), and a former grad student of Harvey's named Eduard Antonyan. It spawned an NPR piece for good measure. And I gathered all the material cut from the article into a massive blog post, which dealt explicitly with this question of whether poker is a game of chance or skill.

If poker is a game of chance, and hence gambling, why do physicists love it so much? Physicists hate to gamble. 'I don't like gambling at all,' Antonyan told me. 'I don't enjoy it and there's nothing in it for me to compensate for the clear negative EV decision of gambling.'

Harvey's not a fan, either: 'Personally I don't like to gamble on games where the house has the odds, but I'm not critical of people who do.' And while the Time Lord gamely learned craps with me while I was writing The Calculus Diaries (it was research, people!), he hasn't been tempted to play craps since.

Binger doesn't mind gambling, per se, but he learned the pitfalls of blackjack as an undergraduate, when he wrote a computer program to beat the game through card-counting (or, as the casinos like to call it, 'cheating') for his senior project. Then he tried to put his strategy into practice. He lost a pile of cash playing blackjack on an ill-fated trip to Reno, and was barred from six casinos in one day for card-counting in a desperate attempt to recoup his losses. 'I realized I wasn't going to get rich playing blackjack,' he recalled.

But poker was different: as he studied the game and pondered the underlying mathematics, Binger realized that poker could be a 'beatable game.'

Fundamentally, poker is a game of skill and strategy, not a game of pure chance (although luck plays a role). (UPDATE: For clarification, when you play poker in a casino, you are playing against the other players -- not the house. The casino takes a cut of the pot, but in essence, you are renting the table. So casinos make far less money off of poker than they do off their usual pure games of chance. They make some -- that's their business model -- but if poker weren't so enormously popular, casinos might not host the games at all.)

Vonk has always loved games, but his love for poker rests on the combination of 'math skills' and 'people skills,' as he put it. 'Good poker requires that you make sound game-theoretic decisions but there is still plenty of freedom to try and outsmart your opponents,' he said. 'Other casino games miss that second element. All you can do in blackjack or roulette is make the best possible mathematical decisions, and even then, you will still lose in the long run. I have never been attracted to those games. It's the fact that you play against other people that makes poker so interesting, and that makes it possible to actually be a winner at the game.'

Math skills help, but that's not all it takes to be a poker badass. Binger said the probability and equity calculations and statistical analysis he applies give him an edge in the game. Vonk finds that his post-game analysis of how he played specific hands benefits from his mathematical skills.

But both Vonk and Binger admit that there are also plenty of other players who really don't know much about the underlying math; they have a good feel, or instinct, for how to play the game.

'There are many people who hate math but are great poker players, but there are hardly any players who lack the people reading abilities and still manage to be good poker players,' said Vonk. 'Mathematical knowledge can to a large extent be replaced by intuition and experience. After a player has played a million hands of poker, even if he does not know the math at all, he will have a decent feeling about when it is profitable to draw to a flush and when it is not.'

That said, knowing the math means you can acquire this kind of knowledge much more quickly, and those skills can give an edge in very rare situations that don't often occur in a poker game. 'To be a great player, you need both!' Vonk insisted. Chris 'Jesus' Ferguson is one of the best players in the world, and definitely relies on math and game theory when he plays (his father is a UCLA mathematician, and the two men have written several papers together):

Antonyan estimated that the game of poker is '90% simple math/general strategy, and 10% understanding the dynamics of the table and/or the attitudes of one or more players towards you as they develop.' The math part rests on basic probability theory, and the probabilities of poker are a bit more complicated because there are many more possible combinations of hands -- plus you're working with incomplete information.

Vonk broke down the process to a few basic questions: What cards do I have? What range of cards do I think my opponent has? Given these, what is the probability I will win the hand after all cards have been dealt? And most important: given that probability, will I make money in the long run when I pay the bet? The best one can do, most of the time, is 'make a very broad guess,' he says. Per the Time Lord (blogging way back in 2004):

'Texas Hold 'Em is so popular because it manages to accurately hit the mark between 'enough information to devise a consistently winning strategy' and 'not enough information to do much more than guess.' The charm in such games is that there is no perfect strategy, in the sense that there is no algorithm guaranteed to win in the long run against any other algorithm. The best poker players are able to use different algorithms against different opponents as the situation warrants.'

To get a sense for how the probabilities can play out, consider the following three possible pairs of hole cards:

Ace-7 unsuited

Pair of sixes

Science Behind Poker Rules

Sean posed this question on Cosmic Variance back in 2006: Which hand is most likely to win if you choose to stay in the pot all the way to the showdown, against other pairs of randomly chosen hole cards? The answer took a whole 'nother blog post to delineate.

Mathematically, it depends on the number of opponents. The probability that you will win goes down as the number of opponents goes up, because there are more ways for you to be beaten. Some hands play well against very few opponents, while others play well against many opponents. It all depends on the circumstances.

Against one opponent, the sixes will win 62.8% of the time, versus 57.3% for Ace-7 and 56.2% for Jack-10 suited. Against four opponents, those odds are reversed: Jack-10 suited will win 27.3% of the time, versus 20.7% for Ace-7 and 17.9% for the pair of sixes.

Science Behind Poker Bar

Why does this happen? 'Against only one randomly-chosen pair of hole cards, there is a substantial chance that the sixes won't need to improve; likewise the ace can often come out on top just by itself, so the Ace-7 is second-best,' Sean explained. 'But against four randomly-chosen pairs of hole cards, chances are excellent that someone will improve, and Jack-10 suited has the best chance.'

The probabilistic outcomes change again if we pit these three hands against each other, two at a time. In that case, sixes are slightly more likely to beat Ace-7, and Ace-7 is likely to beat Jack-10 suited, but Jack-10 suited is likely to beat a pair of sixes.

The sixes are the best starting hand all by themselves. For one of the latter two to win, favorable community cards must appear on the flop, turn, or river. The only way for the Ace-7 to beat paired sixes is for either an ace or a seven to turn up -- or, less likely, for just the right combination of four cards to land on the board to make a straight or flush.

Pit those same sixes against Jack-10 suited, and the situation is reversed. In that scenario, there are more ways for Jack-10 suited to improve. The cards are 'connectors,' so there are more possible cards that would give low straights (7-8-9) and high straights (Q-K-A), plus the hole cards are suited, making it much easier to make a flush.

So Jack-10 suited will usually beat a pair of sixes. But it won't usually beat Ace-7 if the ace is of the same suit. For instance, if four more suited cards come up, the Jack-10 suited will have a flush, but the Ace-7 will have a higher flush, and will win the hand.

See? Poker is a very complicated game, even more so once you add in player behavior during the various rounds of betting. If determining the edge and the odds were all it took to succeed at poker, probability theory would suffice, and one could fairly deem it gambling. If it were a purely logical game like chess, it would merely require impressive feats of calculation to determine the winning series of moves.

But poker is a game of limited information, where players must deduce what cards their opponents are likely to have based on their knowledge of the odds and clues from other players' behavior. There may not be a single answer. As Harvey put it in the Discover article: 'Chess is like classical mechanics. Poker is like quantum mechanics. In chess, there is only one right move. In poker, there is a probability distribution of right moves.'

Harvey admitted that one of his classic errors is 'calling when I think I am beat for other reasons (betting patterns, tells),' but he calls anyway because 'the math says I should. At times like that, I need to pay less attention to the math.'

Human beings aren't always predictably rational, particularly when it comes to poker: if you assume your opponent is skilled and rational, and he isn't, your strategy could backfire and fall victim to 'beginner's luck.'

I found this enlightening analysis over at Cardplayer.com, outlining the different between an optimal strategy and an exploitive strategy (Ferguson's favorite) in No-Limit Texas Hold 'Em. (Note the very specific circumstances described throughout: change even one element and it might call for a different strategy.)

'To take maximum advantage of this terrible opponent, you need to employ an exploitive strategy. The optimal strategy would still win you money but against bad players, other strategies might win you more money. .. An optimal strategy is designed to protect you against opponents who play well. But when we can find ways to do better than optimal strategy against certain players, we do it.'

The article also mentions mathematical/computational giant John von Neumann, who with Oskar Morgenstern (an economist) wrote the definitive treatise on game theory and poker in 1944: Theory of Games and Economic Behavior. It offered an intriguing insight into the art of the bluff: you should always bluff with your worst hand, not a mediocre 'bubble' hand.

If betting is slow, it might be worth calling, or 'limping' into the game, with a mediocre hand, because your chances of winning are pretty good against other mediocre hands (assuming someone isn't 'slow-playing' pocket aces). A bad hand won't win unless everyone else folds, so an aggressive raise is the best strategy.

Indeed, there are rare cases where game theory dictates you should fold pocket aces before the flop when playing a tournament. In non-tournament play, the goal is not just to win the hand but to make the most money. In a tournament, you want to outlast your opponents to win it all. That might entail intentionally opting not to maximize your monetary gains on one specific hand to remain competitive in the tournament. You sacrifice short-term gain to achieve the long-term goal.

It doesn't pay to be too consistent, either. I once played poker with a group that included Harvey. I consistently bet when I had a strong hand, checked when I had a 'bubble' hand, and folded when I had a bad hand. So when I scored with pocket aces and nothing but rags (low cards of varying suits) after the flop, I pushed all-in, going heads-up with Harvey.

He had pocket Queens, a strong hand -- unless one's opponent holds pocket aces. It's hard to fold pocket Queens but that's just what Harvey did. He correctly analyzed his chances, based on my all-too-predictable style of play. I won the hand, but didn't win much money because Harvey folded before he'd committed many chips to the pot.

The optimal strategy can also depend on what type of poker is being played: your strategies will be different for No-Limit Texas Hold-Em, for a No-Limit tournament, a Limit Texas Hold 'Em 'ring game,' and different again for online poker. 'The only people playing online are serious,' said Harvey. 'They also use software to keep track of their opponents' statistics, which is consistent with the rules of the site, even if it seems a bit like cheating.'

Here's the difference: in a live game, you have to remember/keep track of opponents' style of play yourself, i.e, when they raise and in what position. The online software can analyze thousands of hands being played at the same time, and that larger sample space makes for a more accurate statistical analysis.

'It's much more about modeling, statistical analysis and game theory at that level,' said Harvey. 'I'd have to spend as much time learning and playing poker as I do on physics.' And to date, he's been unwilling to do that, unlike Binger, who left physics after scoring big in the 2006 World Series of Poker. He won $4 million with his third place finish, and more than $2 million since. (You can follow his exploits on Twitter: @mwbinger.)

The mathematicians have had a good run when it comes to analyzing poker, but the Time Lord is (rather cheekily) on record predicting that physicists will prove to be the better poker players in the future. His reasoning? No-Limit Texas Hold 'Em is such a complex system that 'we cannot derive a dominant strategy in a closed form.'

'Game theorists and mathematicians study simplified systems about which they can actually prove theorems,' he explained. This is a decent strategy for two players going heads-up, but for a full table, pre-flop, 'it becomes a question of which approximations to make and which models to choose for your opponents.' Physicists, let's face it, are often pretty adept at choosing the best models.

He also had a corollary: 'Phenomenologists and astrophysicists will be better poker players than string theorists.' Take that, Jeff Harvey!

There's a saying that Texas Hold ‘Em consists of long stretches of boredom punctuated by three minutes of sheer terror. Poker never lacks for suspense: you can play a hand flawlessly from a probabilistic standpoint, but there is still the possibility you'll lose; statistical anomalies do happen. In poker, they're known as 'bad beats.' Even with pocket aces and a flop of 9-9-2, your chance of winning a heads-up showdown against pocket queens is only 92%. Harvey once faced just that scenario – and a third queen appeared as the very last card. His opponent 'sucked out on the river.'

Temperament matters too. Poker requires nerves of steel, and an emotional equilibrium that Harvey, for one, admitted he does not possess. 'You need to be unflappable. Bad luck can't bother you. It's too easy to get ‘tilted,' and start playing looser, more erratic, or too passive.' More often than not, he said, 'My emotions get the better of me.'

How can you possibly take into account all those confounding factors in a small study involving 300 players and 60 hands of poker, where the deal is fixed? Quite frankly, you can't. As Dedonno and Detterman concluded in their 2008 paper, 'The reason that poker appears to be a game of luck is that the reliability of any short session is low.. Casino tropez bonus codes. [O]btaining accurate estimates of poker ability may not be easy. Luck (random factors) disguises the fact that poker is a game of skill. However.. skill is the determining factor in long-term outcome.'

Having a good poker face won't hurt your chances either.

Science Behind Poker Game

Adapted from an October 2010 blog post from the archived Cocktail Party Physics blog.

References:

Dedonno, M.A. and Detterman, D.K. (2008) 'Poker Is a Skill,' Gaming Law Review 12(1).

Binger doesn't mind gambling, per se, but he learned the pitfalls of blackjack as an undergraduate, when he wrote a computer program to beat the game through card-counting (or, as the casinos like to call it, 'cheating') for his senior project. Then he tried to put his strategy into practice. He lost a pile of cash playing blackjack on an ill-fated trip to Reno, and was barred from six casinos in one day for card-counting in a desperate attempt to recoup his losses. 'I realized I wasn't going to get rich playing blackjack,' he recalled.

But poker was different: as he studied the game and pondered the underlying mathematics, Binger realized that poker could be a 'beatable game.'

Fundamentally, poker is a game of skill and strategy, not a game of pure chance (although luck plays a role). (UPDATE: For clarification, when you play poker in a casino, you are playing against the other players -- not the house. The casino takes a cut of the pot, but in essence, you are renting the table. So casinos make far less money off of poker than they do off their usual pure games of chance. They make some -- that's their business model -- but if poker weren't so enormously popular, casinos might not host the games at all.)

Vonk has always loved games, but his love for poker rests on the combination of 'math skills' and 'people skills,' as he put it. 'Good poker requires that you make sound game-theoretic decisions but there is still plenty of freedom to try and outsmart your opponents,' he said. 'Other casino games miss that second element. All you can do in blackjack or roulette is make the best possible mathematical decisions, and even then, you will still lose in the long run. I have never been attracted to those games. It's the fact that you play against other people that makes poker so interesting, and that makes it possible to actually be a winner at the game.'

Math skills help, but that's not all it takes to be a poker badass. Binger said the probability and equity calculations and statistical analysis he applies give him an edge in the game. Vonk finds that his post-game analysis of how he played specific hands benefits from his mathematical skills.

But both Vonk and Binger admit that there are also plenty of other players who really don't know much about the underlying math; they have a good feel, or instinct, for how to play the game.

'There are many people who hate math but are great poker players, but there are hardly any players who lack the people reading abilities and still manage to be good poker players,' said Vonk. 'Mathematical knowledge can to a large extent be replaced by intuition and experience. After a player has played a million hands of poker, even if he does not know the math at all, he will have a decent feeling about when it is profitable to draw to a flush and when it is not.'

That said, knowing the math means you can acquire this kind of knowledge much more quickly, and those skills can give an edge in very rare situations that don't often occur in a poker game. 'To be a great player, you need both!' Vonk insisted. Chris 'Jesus' Ferguson is one of the best players in the world, and definitely relies on math and game theory when he plays (his father is a UCLA mathematician, and the two men have written several papers together):

Antonyan estimated that the game of poker is '90% simple math/general strategy, and 10% understanding the dynamics of the table and/or the attitudes of one or more players towards you as they develop.' The math part rests on basic probability theory, and the probabilities of poker are a bit more complicated because there are many more possible combinations of hands -- plus you're working with incomplete information.

Vonk broke down the process to a few basic questions: What cards do I have? What range of cards do I think my opponent has? Given these, what is the probability I will win the hand after all cards have been dealt? And most important: given that probability, will I make money in the long run when I pay the bet? The best one can do, most of the time, is 'make a very broad guess,' he says. Per the Time Lord (blogging way back in 2004):

'Texas Hold 'Em is so popular because it manages to accurately hit the mark between 'enough information to devise a consistently winning strategy' and 'not enough information to do much more than guess.' The charm in such games is that there is no perfect strategy, in the sense that there is no algorithm guaranteed to win in the long run against any other algorithm. The best poker players are able to use different algorithms against different opponents as the situation warrants.'

To get a sense for how the probabilities can play out, consider the following three possible pairs of hole cards:

Ace-7 unsuited

Pair of sixes

Science Behind Poker Rules

Sean posed this question on Cosmic Variance back in 2006: Which hand is most likely to win if you choose to stay in the pot all the way to the showdown, against other pairs of randomly chosen hole cards? The answer took a whole 'nother blog post to delineate.

Mathematically, it depends on the number of opponents. The probability that you will win goes down as the number of opponents goes up, because there are more ways for you to be beaten. Some hands play well against very few opponents, while others play well against many opponents. It all depends on the circumstances.

Against one opponent, the sixes will win 62.8% of the time, versus 57.3% for Ace-7 and 56.2% for Jack-10 suited. Against four opponents, those odds are reversed: Jack-10 suited will win 27.3% of the time, versus 20.7% for Ace-7 and 17.9% for the pair of sixes.

Science Behind Poker Bar

Why does this happen? 'Against only one randomly-chosen pair of hole cards, there is a substantial chance that the sixes won't need to improve; likewise the ace can often come out on top just by itself, so the Ace-7 is second-best,' Sean explained. 'But against four randomly-chosen pairs of hole cards, chances are excellent that someone will improve, and Jack-10 suited has the best chance.'

The probabilistic outcomes change again if we pit these three hands against each other, two at a time. In that case, sixes are slightly more likely to beat Ace-7, and Ace-7 is likely to beat Jack-10 suited, but Jack-10 suited is likely to beat a pair of sixes.

The sixes are the best starting hand all by themselves. For one of the latter two to win, favorable community cards must appear on the flop, turn, or river. The only way for the Ace-7 to beat paired sixes is for either an ace or a seven to turn up -- or, less likely, for just the right combination of four cards to land on the board to make a straight or flush.

Pit those same sixes against Jack-10 suited, and the situation is reversed. In that scenario, there are more ways for Jack-10 suited to improve. The cards are 'connectors,' so there are more possible cards that would give low straights (7-8-9) and high straights (Q-K-A), plus the hole cards are suited, making it much easier to make a flush.

So Jack-10 suited will usually beat a pair of sixes. But it won't usually beat Ace-7 if the ace is of the same suit. For instance, if four more suited cards come up, the Jack-10 suited will have a flush, but the Ace-7 will have a higher flush, and will win the hand.

See? Poker is a very complicated game, even more so once you add in player behavior during the various rounds of betting. If determining the edge and the odds were all it took to succeed at poker, probability theory would suffice, and one could fairly deem it gambling. If it were a purely logical game like chess, it would merely require impressive feats of calculation to determine the winning series of moves.

But poker is a game of limited information, where players must deduce what cards their opponents are likely to have based on their knowledge of the odds and clues from other players' behavior. There may not be a single answer. As Harvey put it in the Discover article: 'Chess is like classical mechanics. Poker is like quantum mechanics. In chess, there is only one right move. In poker, there is a probability distribution of right moves.'

Harvey admitted that one of his classic errors is 'calling when I think I am beat for other reasons (betting patterns, tells),' but he calls anyway because 'the math says I should. At times like that, I need to pay less attention to the math.'

Human beings aren't always predictably rational, particularly when it comes to poker: if you assume your opponent is skilled and rational, and he isn't, your strategy could backfire and fall victim to 'beginner's luck.'

I found this enlightening analysis over at Cardplayer.com, outlining the different between an optimal strategy and an exploitive strategy (Ferguson's favorite) in No-Limit Texas Hold 'Em. (Note the very specific circumstances described throughout: change even one element and it might call for a different strategy.)

'To take maximum advantage of this terrible opponent, you need to employ an exploitive strategy. The optimal strategy would still win you money but against bad players, other strategies might win you more money. .. An optimal strategy is designed to protect you against opponents who play well. But when we can find ways to do better than optimal strategy against certain players, we do it.'

The article also mentions mathematical/computational giant John von Neumann, who with Oskar Morgenstern (an economist) wrote the definitive treatise on game theory and poker in 1944: Theory of Games and Economic Behavior. It offered an intriguing insight into the art of the bluff: you should always bluff with your worst hand, not a mediocre 'bubble' hand.

If betting is slow, it might be worth calling, or 'limping' into the game, with a mediocre hand, because your chances of winning are pretty good against other mediocre hands (assuming someone isn't 'slow-playing' pocket aces). A bad hand won't win unless everyone else folds, so an aggressive raise is the best strategy.

Indeed, there are rare cases where game theory dictates you should fold pocket aces before the flop when playing a tournament. In non-tournament play, the goal is not just to win the hand but to make the most money. In a tournament, you want to outlast your opponents to win it all. That might entail intentionally opting not to maximize your monetary gains on one specific hand to remain competitive in the tournament. You sacrifice short-term gain to achieve the long-term goal.

It doesn't pay to be too consistent, either. I once played poker with a group that included Harvey. I consistently bet when I had a strong hand, checked when I had a 'bubble' hand, and folded when I had a bad hand. So when I scored with pocket aces and nothing but rags (low cards of varying suits) after the flop, I pushed all-in, going heads-up with Harvey.

He had pocket Queens, a strong hand -- unless one's opponent holds pocket aces. It's hard to fold pocket Queens but that's just what Harvey did. He correctly analyzed his chances, based on my all-too-predictable style of play. I won the hand, but didn't win much money because Harvey folded before he'd committed many chips to the pot.

The optimal strategy can also depend on what type of poker is being played: your strategies will be different for No-Limit Texas Hold-Em, for a No-Limit tournament, a Limit Texas Hold 'Em 'ring game,' and different again for online poker. 'The only people playing online are serious,' said Harvey. 'They also use software to keep track of their opponents' statistics, which is consistent with the rules of the site, even if it seems a bit like cheating.'

Here's the difference: in a live game, you have to remember/keep track of opponents' style of play yourself, i.e, when they raise and in what position. The online software can analyze thousands of hands being played at the same time, and that larger sample space makes for a more accurate statistical analysis.

'It's much more about modeling, statistical analysis and game theory at that level,' said Harvey. 'I'd have to spend as much time learning and playing poker as I do on physics.' And to date, he's been unwilling to do that, unlike Binger, who left physics after scoring big in the 2006 World Series of Poker. He won $4 million with his third place finish, and more than $2 million since. (You can follow his exploits on Twitter: @mwbinger.)

The mathematicians have had a good run when it comes to analyzing poker, but the Time Lord is (rather cheekily) on record predicting that physicists will prove to be the better poker players in the future. His reasoning? No-Limit Texas Hold 'Em is such a complex system that 'we cannot derive a dominant strategy in a closed form.'

'Game theorists and mathematicians study simplified systems about which they can actually prove theorems,' he explained. This is a decent strategy for two players going heads-up, but for a full table, pre-flop, 'it becomes a question of which approximations to make and which models to choose for your opponents.' Physicists, let's face it, are often pretty adept at choosing the best models.

He also had a corollary: 'Phenomenologists and astrophysicists will be better poker players than string theorists.' Take that, Jeff Harvey!

There's a saying that Texas Hold ‘Em consists of long stretches of boredom punctuated by three minutes of sheer terror. Poker never lacks for suspense: you can play a hand flawlessly from a probabilistic standpoint, but there is still the possibility you'll lose; statistical anomalies do happen. In poker, they're known as 'bad beats.' Even with pocket aces and a flop of 9-9-2, your chance of winning a heads-up showdown against pocket queens is only 92%. Harvey once faced just that scenario – and a third queen appeared as the very last card. His opponent 'sucked out on the river.'

Temperament matters too. Poker requires nerves of steel, and an emotional equilibrium that Harvey, for one, admitted he does not possess. 'You need to be unflappable. Bad luck can't bother you. It's too easy to get ‘tilted,' and start playing looser, more erratic, or too passive.' More often than not, he said, 'My emotions get the better of me.'

How can you possibly take into account all those confounding factors in a small study involving 300 players and 60 hands of poker, where the deal is fixed? Quite frankly, you can't. As Dedonno and Detterman concluded in their 2008 paper, 'The reason that poker appears to be a game of luck is that the reliability of any short session is low.. Casino tropez bonus codes. [O]btaining accurate estimates of poker ability may not be easy. Luck (random factors) disguises the fact that poker is a game of skill. However.. skill is the determining factor in long-term outcome.'

Having a good poker face won't hurt your chances either.

Science Behind Poker Game

Adapted from an October 2010 blog post from the archived Cocktail Party Physics blog.

References:

Dedonno, M.A. and Detterman, D.K. (2008) 'Poker Is a Skill,' Gaming Law Review 12(1).

Ferguson, Thomas, and Ferguson, Chris. (2003) 'On the Borel and von Neumann Poker Models,' Game Theory and Applications 9 (2003), 17-32.

Ferguson, Thomas, and Ferguson, Chris. (2007) 'The Endgame in Poker,' Optimal Play - Mathematical Studies of Games and Gambling, 79-106. Stewart Ethier and William R. Eadington, eds. Reno, NV: Institute for the Study of Gambling and Commercial Gaming.

Science Behind Poker Games

Ferguson, Thomas, Ferguson, Chris, and Garwargy, Cephas. (2007) 'U(0,1) Two-Person Poker Models,' Game Theory and Applications 12, 17-37.

Fiedler, Ingo C. and Rock, Jan-Philipp. (2009) 'Quantifying Skill in Games—Theory and Empirical Evidence for Poker,' Gaming Law Review and Economics 13(1): 50-57.

Meyer, G., von Meduna, M., Brosowski, T., and Hayer, T. (2012) 'Is Poker a Game of Skill or Chance? A Quasi-Experimental Study,' Journal of Gambling Studies, Online First, August 15, 2012.

von Neumann, John and Morgenstern, Oskar. Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press, 1944.