When Confronted with An Explanation that is Based on Mathematics

[sfer]

Because we're naturally inclined to learn experientially, and building models is like the opposite way to learn. It's why bad poker players make terrible plays for years, like not value betting the river enough in HE. They remember the times they get raised/checkraised rather than thinking mathematically about what exactly they're risking and how often they have to be right.

Also, explaining modern portfolio theory to someone who is disinclined to want to listen is the most frustrating thing on earth.

[JoshuaD]

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You can't stick all the time with the math either, because pure math will get you nonsensical results like .9999999 = 1.0, in that old puzzle of:

1/9 = .111111111111111111111111111111111
x9
9/9 = .999999999999999999999999999999999

9/9 = 1.0

1.0 = .999999999999999999999999999999999

But, back to the point, the problem is that people are mathematically illiterate in many respects and explaining things in such a fashion will only get glassy-eyed looks of ignorance.

Such is life.


Barron Vangor Toth
www.BarronVangorToth.com

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0.9999... (repeating forever) does (exactly) equal 1.

It's just another way of expressing the same number.

It's the same way 2/5 = 4/10. Just because they don't look the same, doesn't mean they aren't the same.

I really don't feel like typing out the mathematical proof (mainly because I can't exactly remember it), but they are EXACTLY the same number. "0.9999..." is just a ugly result of the decimal number system.


Edit: If that doesn't convince you, look at it from this perspective.

Two numbers, A and B, are said to be different when tehre exists some X such that, A < X < B.

Said more simple, two numbers are the same if there's a number between them (incidently, in the real number system if there is one number, there's an infinte number).

It should be easy to see that no such number exists in the case of .999... and 1, and therefore they're the same number.

[timmer]

Ockhams Razor is why.

[chezlaw]

Lots of people dont like thinking and maths requires thinking. Lots of other people are scared of maths.

or maybe he was just deeply uninterested in portfolio theory.

chez

[MicroBob]

It's not just those that are 'generally' poor at math though.
Some who are, in fact, good in certain areas of math still fail miserably in other areas and continue to trust their own flawed instincts or self-delusional observations.


I know a guy who is quite good at general math. Can multiply a couple of 4-digit numbers in his head (which is something I can't do)....but he swears that roulette is beatable by varying one's bet and by noticing certain 'patterns'. he was convinced that 32 was more likely to fall if the previous spin was 23....and vice-versa. and that if black fell 3x in a row it was better than 50% to fall again on the next spin.
Again, if you asked him "Hey jerry. What's 7629 X 2564?" he would give you the correct answer.


My sister's boyfriend took several math and statistics courses in college....and swears that he has a winning system for craps because he always leaves after he has won $100 or lost $50...whichever happens first.
"see?? that way when I win I am winning more than when I lose."
Again, he got A's in his college courses. But he uses this 'system' because some pit-boss friend of his in LV taught it to him.
I am not quite articulate enough to entirely convince him that the number of times he loses only $50 will happen more than the times he wins $100 to cause net-losses in the long-run.
but I did start to get somewhere by explaining that "on each roll...if you are at a 1% disadvantage...then you remain at a 1% disadvantage no matter how long you play."



More to your original point however....when I dealt BJ (for 8 mths) I very rarely had a player who actually played with the correct basic-strategy (some would come close...but then fail to double their soft-18 or split their 9's or some other kind of play that most non-expert players don't know to do).
Many of these players would stand on their 14-16 vs. 7-10 and say "I don't care what the book says. Hitting that just doesn't work for me."
If someone else told them that the book is correct and it was developed by math people smarter than us who plugged millions of hands into a computer to determine what plays would give the player the best chance then they would argue "But that's just stupid. I don't care what the math says or what 'the book' says. You can just tell that if you hit a 15 against a 7 you are going to lose. If you can't see that then you're just stupid. Why on earth would you play 'by the book' when the book is so obviously wrong?"
I have dealt to higher-income professionals (doctors, lawyers, etc) who subscribed to this logic.


I have often wondered why I am smart enough to know that the MIT math-types are better for BJ strategy even on plays that normally would be counter-intuitive to my own instincts. I have the ability to ignore my instincts and make the play the the 'experts' tell me is correct because I know they are smarter than me.

I agree that it's interesting stuff though.

[Paul2432]

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One example of this is the "hot hand" in basketball. Statistical analysis has shown that this is a myth. That is a particular player has exactly the same chance of making his next shot whether he has missed his last five shots or made his last five shots. Every discussion I have seen of this topic has resulted in vigorous rebuttal from basketball players. The rebuttal is always based on personal experience and never on the actual statistical analysis.

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Not really on the subject of mathamatical debating, but I disagree with your belief that the "hot hand", as you call it, is entirely mythical. There is such a thing as a player getting into his rhytem. It probably has to do with muscle memory or something similar. I agree that it's not because he is just "on fire tonight" or some other made up explaination, but science and human anatomy can prove that muscle memory is very real, and could very well affect the performance of a basketball player.

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Intuition has its place, and mathematical analyses have their place. Neither is the absolutely correct way to make a decision. Paul's example of the 'hot hand' myth I feel is incorrect; how is mathematical analysis going to take into account all of the factors that go into making a shot? Of course it isn't going to do that, it only looks at the results. Such things have little basis when we're looking at one event. The theory of probability and statistical analysis are based on ceteris paribus (all else being equal). Since we know that isn't the case in the "hot hand" example, our intuitions of human psychology may do a hell of a lot better than statistical analysis.

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I knew when I posted this I would get these kinds of responses. Like I predicted people are rebutting based on personal experience and not refuting the statistical analysis. One of the points of the research is that people can believe things very strongly that simply are not true. I had hoped that poker players would understand the nature of streaks and how they can lead a person to conclusions that do not stand up to mathematical scrutiny.

Just to be clear, this is not my theory. When I first heard it I was incredulous myself. However, this has been studied many times over the last twenty years by professors of mathematics and psychology. The findings have been published and scrutinized. Every single time the finding is the same: there is no such thing as a hot hand. It is purely a psychological phenomonon.

A google search reveals dozens of links on the subject. Here are three:

More recent hot hand study
clustering web site with section on hot hands
web site devoted to hot hand research

To Adios (original poster):

You wrote:

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I'm fairly certain that most people eschew explanations that are based on mathematical models. . . . , " I don't know about the math you're mentioning but common sense tells me this ..."

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It looks like you were right on.

Paul

[Triumph36]

Did the statistical analysis take into account every factor? That is to say, the time of the game, the reaction of the other players, and so on? Of course not. So you've proven your point, but I hardly think that it's a myth if everyone believes it. That is, while it may be a myth, the psychological implications of belief in such a myth are sure to impact the on-court play.

That said, would you give the last shot to the player who'd been missing his shots all night? Of course not. There's probably something wrong with him. Statistical analysis would say he's no more likely to miss that shot than he is any other shot, but that is clearly incorrect in this case.

Analysis and the scientific method look great the wider our perspective is. When we try to take into account all the factors, analysis and the scientific method break down. They do a hell of a lot better than intuition but they are highly fallible, especially to those capable of making serious errors in judgment.

This is made evident in No Limit Hold 'Em. Sometimes, players are forced to make decisions based on extremely unlikely holdings for the opponent. Statistically, it is incredibly unlikely that a player is holding AA when we are holding say, two queens. But we are forced by his betting to make decisions based on the now incredibly likely possibility that our opponent is beating us.

My point is, statistical analysis can make plenty of errors, especially when misapplied. I feel that the 'hot hand' myth could easily be misapplied by an 'academic' coach who gives the ball to a guy who obviously doesn't have it that night. I'm simply not ready to worship at the altar of the scientific method. Plus poker players like Doyle Brunson make me think that intuition can do a lot better than analysis in a rare few.

[Lawrence Ng]

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It's not just those that are 'generally' poor at math though.
Some who are, in fact, good in certain areas of math still fail miserably in other areas and continue to trust their own flawed instincts or self-delusional observations.

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People have heavily relied upon all their life based on fallacies, myths, biases, and karma to the extent that scientific reasoning is still a disbelief to them.

I have a friend like you who can spit complex calculations as easy and as fast as you and I could punch them into a calculator. However, put him in a poker game and he'll say his favorite hand is 6-9 (for obvious reasons) and he'll always raise with it. This, mathematically and statistically, makes absolutely no sense and is for the most part a definite -EV play.

The nature of gambling takes on one where it leads players into heavy supersitition. This is especially true in most Asian cultures where people consider it extremely bad luck if you tap them on the shoulders or stand too close them when they are gambling.

Casino's use this pyschological tactic well to their advantage. Major casinos contruct and design the game floor to visually mesmerize players. Once players start playing with the bright lights, action and sounds they get semi-hypnotized. Some players will find it hard to get up and leave and the design of the large casinos makes it so that walking out that door is hard and a long walk sometimes.

Sure enough, in most house games where the rules and definitions are fixed it is not possible to beat the games in the long run. Having said that, one should never play them apart from entertainment value purposes. So I guess from that standpoint, it's ok to have some fun and "go with the flow/hot hand" situation. However, failing to realize that the previous result is completely independent and exclusive to the next result is detrimental indeed.

Lawrence

[TomCollins]

One of the biggest mistakes people who are fairly good at analyzing situations make is failing to commpare it to an alternative.

For instance, I could say "Is good to win $100?" But if I say, the alternative is a 1/5 chance at winning $10,000, its very clear. Identifying that there often is an alternative is something most people overlook.

In the lesser of two evils, most people are reluctant to choose a "bad" choice, even if not acting results in a worse choice. That's exactly what hitting on a 16 is. You will usually lose. But you are going to usually lose anyway.

[Rudbaeck]

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Edit: If that doesn't convince you, look at it from this perspective.

Two numbers, A and B, are said to be different when tehre exists some X such that, A < X < B.

Said more simple, two numbers are the same if there's a number between them (incidently, in the real number system if there is one number, there's an infinte number).

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Thank you for saving me half an hour in a dusty attic looking for half forgotten textbooks!