New article: Runs & Gaps
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#5
Bayes' Blog / The Long Run
May 10, 2016, 03:12:28 PM
New article: http://www.roulettician.com/articles/article2.html
Apologies to anyone who signed up for email subscriptions. It doesn't seem to be working, but I'm looking into it.
Apologies to anyone who signed up for email subscriptions. It doesn't seem to be working, but I'm looking into it.
#6
Bayes' Blog / Web Site
March 12, 2015, 01:51:55 PM
Here is some shameless self-promotion. 
I mentioned a web site I'm working on a few days ago. I've since decided to widen the scope of it, so instead of it being mainly about my "system" and how to play it, it's now going to cover much more general stuff about probability, stats, and how to design (even-money) systems.
I will still be giving details of how I play and making the software available, but that will only be a relatively small part of what's on the site. When I've finished the core sections I'll be adding blog posts regularly on related topics.
This is a long-term project and I've only just started, so check back every week or so for updates.
So for anyone who's been reading it, I've made some changes, particular in the introduction.
http://www.roulettician.com
cheers!
I mentioned a web site I'm working on a few days ago. I've since decided to widen the scope of it, so instead of it being mainly about my "system" and how to play it, it's now going to cover much more general stuff about probability, stats, and how to design (even-money) systems.
I will still be giving details of how I play and making the software available, but that will only be a relatively small part of what's on the site. When I've finished the core sections I'll be adding blog posts regularly on related topics.
This is a long-term project and I've only just started, so check back every week or so for updates.
So for anyone who's been reading it, I've made some changes, particular in the introduction.
http://www.roulettician.com
cheers!
#7
Bayes' Blog / Preamble to The Prediction Formula
October 23, 2014, 03:17:22 PM
As a preamble to the formula itself, I'd like to say something about the accepted viewpoint regarding games of so-called "pure chance", such as roulette, craps, and baccarat.
Everyone "knows" that, for example, R/B has a 50:50 chance of hitting (ignoring the zero). But it's important to understand that this depends on a certain interpretation of probability. This interpretation is based on relative frequency: that in an infinite series of trials the ratio of reds to blacks will approach 1. In practice, the ratio approximates to 1 in only a few hundred trials or less in most cases, but even so, the definition of probability as a relative frequency means that it makes no sense to predict future outcomes based on past outcomes, because by definition the probability of red is fixed at 1/2, (the ratio of reds to [reds + blacks] in an infinite number of trials) so you can only ever "predict" the past, not the future.
Another interpretation of probability involves symmetry. Thus, it seems intuitively obvious that if there are X ways that we can get a certain outcome (say Red) and Y ways we can get another outcome (Black), and neither of these outcomes seems more likely than the other, then the probability of the outcome is again fixed at 1/2, which is the ratio of the number of ways the event of interest can happen to the total number of possibilities. This is called the Classical interpretation, and is the one most often used in elementary applications. It was the first notion in our understanding of probability (which was developed in the context of games of chance), and is most appropriate in those situations where the symmetry is obvious- each outcome has (so it seems) an equal chance of occurring, so in the case of N equally likely outcomes, each has a probability of 1/N (e.g. each outcome in roulette has a probability of 1/37)
Again, this interpretation does not allow any predictions of future outcomes because the symmetry is "built into" the game (it's assumed), and since trials are independent, on every trial there are just as many ways red can occur as black, so the probability of Red is again fixed in stone at 1/2, and accordingly, it makes no sense to predict the future (you can only predict the past).
Notice that both these interpretations assume that the idea of "randomness" is something built into the system in question. e.g. Roulette outcomes are "random" because of the symmetry of the wheel; dice outcomes are similarly random because (again, by assumption), no side is more likely to land uppermost than any other. Furthermore, we can do some physical experiments to confirm this, so we roll the die a few thousand times and see that indeed, the relative frequency of any side approaches 1/6, just as symmetry considerations suggest that it should.
The key idea I'm trying to get across is that as regards probability, these interpretations (which seem particularly well suited to games like roulette) imply that the probabilities are fixed and somehow objectively a part of the games themselves: the odds of red/black are just 50:50 - end of story. Therefore, bet selection (particularly if based on past results) is meaningless, all systems are useless, and only the casino can win "in the long term".
Now, logically, this is perfectly true, but logic can only tell you what conclusion(s) follow from certain premises, it can say nothing about whether the premises themselves are true.
In fact, there is another interpretation of probability, which includes the others as special cases, and holds that it is subjective, not objective. In a nutshell, probability is in the mind, not something "out there", as an attribute or property of a thing.
That this is obvious doesn't really require much argument. If you have information regarding a certain roulette wheel that I don't have (for example, that it's biased), then your probability of the ball falling into say, pocket 13 may be different from mine. Are either of us wrong, logically speaking? No, it's just that our conclusions are based on difference premises.
It's the same thing with probabilities.Take the action of flipping a coin. Is it part of what it means to be a coin that the probability of it coming up heads if you flip it is 1/2? Not at all; the (normally unexpressed) premise which goes with the statement "the chance of the coin landing heads is 1/2" is that the coin is "fair". But what does "fair" mean? Well, it's assumed that it isn't weighted on one side more than the other (symmetry), but also you have to take physics into account in the way it's flipped. According to Newtonian mechanics, the result of flipping is perfectly deterministic and depends on the initial velocity given to the coin given an initial position. In theory, a precisely made coin flipping machine which gave the same initial force to the coin, which was placed in exactly the same way into the machine every time, would always result in the same outcome.
The "Prediction Formula", which I'll be posting soon, is based on the subjective view of probability, and is derived from Bayes' theorem, which is the key equation used to make inferences about data.
The main purpose of statistics is to come to some conclusions about a "population" based on a sample. For example, sticking with roulette, the sample is some number of "trials" or outcomes, and the population is the total number of outcomes which could be sampled. It's best to regard the roulette system (wheel + ball etc) as an urn full of balls which have the numbers 0-36 written on them. Each trial consists of you taking a ball from the urn, then replacing it. You would like to build up a picture of the "population" from your trials regarding the composition of the balls in the urn. Specifically, you want to do this for purposes of prediction, how do you do it?
more to come...
Everyone "knows" that, for example, R/B has a 50:50 chance of hitting (ignoring the zero). But it's important to understand that this depends on a certain interpretation of probability. This interpretation is based on relative frequency: that in an infinite series of trials the ratio of reds to blacks will approach 1. In practice, the ratio approximates to 1 in only a few hundred trials or less in most cases, but even so, the definition of probability as a relative frequency means that it makes no sense to predict future outcomes based on past outcomes, because by definition the probability of red is fixed at 1/2, (the ratio of reds to [reds + blacks] in an infinite number of trials) so you can only ever "predict" the past, not the future.
Another interpretation of probability involves symmetry. Thus, it seems intuitively obvious that if there are X ways that we can get a certain outcome (say Red) and Y ways we can get another outcome (Black), and neither of these outcomes seems more likely than the other, then the probability of the outcome is again fixed at 1/2, which is the ratio of the number of ways the event of interest can happen to the total number of possibilities. This is called the Classical interpretation, and is the one most often used in elementary applications. It was the first notion in our understanding of probability (which was developed in the context of games of chance), and is most appropriate in those situations where the symmetry is obvious- each outcome has (so it seems) an equal chance of occurring, so in the case of N equally likely outcomes, each has a probability of 1/N (e.g. each outcome in roulette has a probability of 1/37)
Again, this interpretation does not allow any predictions of future outcomes because the symmetry is "built into" the game (it's assumed), and since trials are independent, on every trial there are just as many ways red can occur as black, so the probability of Red is again fixed in stone at 1/2, and accordingly, it makes no sense to predict the future (you can only predict the past).
Notice that both these interpretations assume that the idea of "randomness" is something built into the system in question. e.g. Roulette outcomes are "random" because of the symmetry of the wheel; dice outcomes are similarly random because (again, by assumption), no side is more likely to land uppermost than any other. Furthermore, we can do some physical experiments to confirm this, so we roll the die a few thousand times and see that indeed, the relative frequency of any side approaches 1/6, just as symmetry considerations suggest that it should.
The key idea I'm trying to get across is that as regards probability, these interpretations (which seem particularly well suited to games like roulette) imply that the probabilities are fixed and somehow objectively a part of the games themselves: the odds of red/black are just 50:50 - end of story. Therefore, bet selection (particularly if based on past results) is meaningless, all systems are useless, and only the casino can win "in the long term".
Now, logically, this is perfectly true, but logic can only tell you what conclusion(s) follow from certain premises, it can say nothing about whether the premises themselves are true.
In fact, there is another interpretation of probability, which includes the others as special cases, and holds that it is subjective, not objective. In a nutshell, probability is in the mind, not something "out there", as an attribute or property of a thing.
That this is obvious doesn't really require much argument. If you have information regarding a certain roulette wheel that I don't have (for example, that it's biased), then your probability of the ball falling into say, pocket 13 may be different from mine. Are either of us wrong, logically speaking? No, it's just that our conclusions are based on difference premises.
It's the same thing with probabilities.Take the action of flipping a coin. Is it part of what it means to be a coin that the probability of it coming up heads if you flip it is 1/2? Not at all; the (normally unexpressed) premise which goes with the statement "the chance of the coin landing heads is 1/2" is that the coin is "fair". But what does "fair" mean? Well, it's assumed that it isn't weighted on one side more than the other (symmetry), but also you have to take physics into account in the way it's flipped. According to Newtonian mechanics, the result of flipping is perfectly deterministic and depends on the initial velocity given to the coin given an initial position. In theory, a precisely made coin flipping machine which gave the same initial force to the coin, which was placed in exactly the same way into the machine every time, would always result in the same outcome.
The "Prediction Formula", which I'll be posting soon, is based on the subjective view of probability, and is derived from Bayes' theorem, which is the key equation used to make inferences about data.
The main purpose of statistics is to come to some conclusions about a "population" based on a sample. For example, sticking with roulette, the sample is some number of "trials" or outcomes, and the population is the total number of outcomes which could be sampled. It's best to regard the roulette system (wheel + ball etc) as an urn full of balls which have the numbers 0-36 written on them. Each trial consists of you taking a ball from the urn, then replacing it. You would like to build up a picture of the "population" from your trials regarding the composition of the balls in the urn. Specifically, you want to do this for purposes of prediction, how do you do it?
more to come...
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