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On Probability

reason

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The probability of rolling a 'five' with an ordinary die is 1/6. Therefore, it follows that the proposition 'the 3rd roll of an ordinary die will be a "five"' has a 16% probability of being true, right? Well, not exactly. It may seem quite a reasonable conclusion, but there a problem lurking away from plain sight. The purpose of this thread is to bring that problem into the light.

If the probability of rolling a 'five' with an ordinary die is 1/6 then it would be sensible to predict 'five' to turn up once for every six rolls of the die. However, if the proposition 'the 3rd roll of an ordinary die will be a "five"' has a 16% probability of being true, is it sensible to predict that the same 6th roll of the die will turn up a 'five' once for every six same 6th rolls of the die? That is a contradiction. The same six throws would not be the same if only one turned up a six, and the only way to resolve the contradiction is if all of the six same throws turn up a 'five' or some other number. Therefore, the proposition 'the 3rd roll of an ordinary die will be a "five"' can only have a probability of either 1 or 0. In other words, propositions are never probably true.

For example, imagine that someone was tasked with desgining a fleet of aircraft. That person surveyed the scientific literature and discovered the theory of gravity. However, he also noted a consensus between scientists that this theory is only probably true, about 80% probable. Thinking on this the aircraft designer came to a decision: he would design his aircraft so that they would not only function correctly according the theory of gravity, but also function correctly according to an alternative, less probable, theory of gravity. Afterall, would it not be sensible to predict that the prevailing theory of gravity will be false for two of every ten flights?

That said, after more scientific evidence makes the theory more probable, perhaps 90%, it would be sensible to only expect the theory to be false for one of every ten flights, right?

The strange thing is that improbable theories are actually the best. The most probable theories are tautological, and utterly useless. For example, consider again the ordinary die that is to be rolled, and the following two predictions.

P1: the next roll will be a one.*
P2: the next roll will be a one, two, or three.​

Now which of these predictions is the most probable? Well, the probability of P1 is 1/6, whereas the probability of P2 is 1/2. In short, P2 is more probable, but it is also the least informative. P1, if true, is much preferrable to P2, since it is far more specific. Moreover, note that improbable theories are not only more informative, but usually more falsifiable i.e. experimentally testable.

Okay, that's enough, my incomprehensible rant is over...

* The word 'next' in the statement 'the next roll will be a one' is a varible, which makes the statement an unfinished proposition i.e. it does not say anything about the world until the 'next' is specified. It is interesting to note that until the variable is satisfied we can sensibly talk about it with probabilities, since we are not actually asserting anything about the world
 

Jack Flak

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Die roll: The probability of a 5 roll on the third throw is always 1/6.

Plane: Theory's either right or wrong to begin with. If they estimate that the theories they're using in airliner construction are 80% likely to be valid, I'm going to stop flying.

Testable: Mathematically, it should be just as easy to test 1/2 as 1/6.

ARE YOU JUST BEING SILLY LEE
 

reason

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Die roll: The probability of a 5 roll on the third throw is always 1/6.
That depends the whether phrase 'third throw' is intended a variable representing the rolling of any possible third throw or as a proposition. If it is a proposition, as specified, then the probability that it is true is either 1 or 0, not 1/6.

Testable: Mathematically, it should be just as easy to test 1/2 as 1/6.
I meant 'empirically testable', 'experimentally testable', or 'scientifically testable'. The prediction that the next roll will turn up a one is falsified by rolling a two, three, four, five or six, whereas the prediction that the next roll will turn up a one, two or three, can only be falsified by rolling a four, five or six. In other words, the former is more testable, less probable, and more informative (i.e. greater information content).
 

Jack Flak

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That depends the whether phrase 'third throw' is intended a variable representing the rolling of any possible third throw or as a proposition. If it is a proposition, as specified, then the probability that it is true is either 1 or 0, not 1/6.

I meant 'empirically testable', 'experimentally testable', or 'scientifically testable'. The prediction that the next roll will turn up a one is falsified by rolling a two, three, four, five or six, whereas the prediction that the next roll will turn up a one, two or three, can only be falsified by rolling a four, five or six. In other words, the former is more testable, less probable, and more informative (i.e. greater information content).
Probability is in reference to the future. Only the result has a 1 or 0 "probability," after the fact.

I can't seem to fathom how the test will be any different. You hypothesize that approx. 1 in 6 will be fives, roll 1000 dice, record results. Or, that 1 in 2 will be (4,5,6), roll 1000 dice, record results. I think you'll find the hypothesis will almost always prove correct.
 

reason

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Probability is in reference to the future. Only the result has a 1 or 0 "probability," after the fact.
No, that doesn't matter. It's too late to discuss it further now and I need to go to sleep, but my basic point is that it makes no sense to say that any proposition (or theory) has a probability other than 1 or 0 (true or false). That does not mean that we cannot talk sensibly about probability, only that it is only with nonpropositional statements i.e. statements which do not make claims about the facts. Of course, sometimes the word 'probability' is used to mean somethign quite different, and in which case what I wrote may not apply.

I can't seem to fathom how the test will be any different. You hypothesize that approx. 1 in 6 will be fives, roll 1000 dice, record results. Or, that 1 in 2 will be (4,5,6), roll 1000 dice, record results. I think you'll find the hypothesis will almost always prove correct.
I think that you have misunderstood. The P1 and P2 predictions were not probabilistic, each asserted that the next roll of the ordinary die will be some number or other, and are false if that number does not come up.
 

Jack Flak

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Perhaps you're referring to theory which has no perceivable relation to reality. Or is it determinism? That the die already "knows" what it's going to show for every roll, the moment it's created? If so, it seems a bit of a fool's errand to analyze it.
 

ygolo

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Regarding the emperical testing of probablistic phenomena

Statistics is the emperical testing of probabilistic models.

One of the problems with the way probability and statistics are present to the lay person is that things are presented as individual number instead of as a distribution.

In terms of a single roll of a "fair" dice, all the outcomes are equally likely (that's what makes the dice fair).

We don't emperically test individual events rulef by probability but the distributions we get upon running many tests.

In fact, in modern science (at least at the leading edge), it is nearly impossible not have statistical or probabilistic thinking as part of the modeling.
 

IlyaK1986

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Blah blah blah I don't understand basic probability and am confusing myself with my own rant.

The probability of any one value on any given roll of a fair multi-sided object is always 1/n, where n is the number of sides. That is because the geometric distribution is memoryless, or in other words, the probability of obtaining a given value on any roll of a fair die is completely independent of what came before (or after) it. That's just common knowledge, and if you don't understand this, then you shouldn't be talking about probability.

Furthermore, you can empirically test the number of fives by conducting a vast amount of trials. A small case of rolls prove nothing. In order to verify something experimentally, you need a vast amount of trials. This is what is called the law of large numbers, and is also part of basic probability.

So you're getting confused and ranting over some basic tenets of probability and vapid semantics of theories? Cut me a break.
 

reason

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So you're getting confused and ranting over some basic tenets of probability and vapid semantics of theories? Cut me a break.
Hmm. No, you do not understand what I mean to say. It's very abstract and unintuitive and I am having a difficult time trying to explain it clearly (which was why I wrote the post, as a test). The objections which you refer to are of no concern to me or my argument, though at least one is mistaken (at least as a matter of logic, though maybe not methodology).
 

reason

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In fact, in modern science (at least at the leading edge), it is nearly impossible not have statistical or probabilistic thinking as part of the modeling.
There is some of what you have written which, though interesting in its own right and worthy of discussion, does not concern my argument. The confusion is, perhaps, of my own making, as my original post was written very late at night about a matter which I am having difficulty discussing anyway. I think that soon I will be in a position to explain more clearly what I mean. The thought needs to gestate for a little while longer.
 

reason

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Here I go again.

The wikipedia article on truth states that there are 'differing claims on [what] constitutes truth' and 'whether truth is subjective, relative, objective, or absolute'. The article goes on explain that there are 'major theories of truth', such as correspondence theory, coherence theory, consensus theory and pragmatic theory, each with its own proponents, and states that 'theories of truth continue to be debated'. But what is being debated? This "debate" is framed as though theories of truth are competing, but are they? I do not think so.

If two scientific theories compete then they are expected to contradict one another, but theories of truth do not. The word 'post' can refer to 'a piece of wood or other material set upright into the ground to serve as a marker or support', 'a starting point at a racetrack', or perhaps 'an electronic message sent to a newsgroup or forum'. It would, however, seem very strange to call these alternatives meanings 'theories of post', and yet that seems to be exactly what has been done in regard to truth.

Suppose that a scientist is interested in finding theories which correspond to the facts, and by convention she labels such theories 'true'. One day a philosopher comes to watch the scientist, and asks the scientist whether she has had any success finding true theories. The scientists is careful not to make any guaruntees, but answers that she has been successful. At this point the philosopher kindly informs the scientist that her theories could not possibly be true, because they have not yet been agreed upon. The scientist would perhaps be confused. Even if we suppose the philosopher later convinced her of the consensus theory of truth, then should the scientist revise the aim of her investigations so to find theories which everyone agrees upon? No, the original aim was the discovery of theories which correspond to the facts, and that aim can remain even if the scientist no longer labels such theories 'true'. Nothing depended upon the meanings of particular words.

If the "theories of truth" are competing in any capacity then it is not in regard to the essential meaning of the word 'true', but as competing proposals for the adoption of a convention. With this in mind the most sensible "theory of truth" is the correspondence theory, because when most people say 'P is true' they mean something like 'P corresponds to the facts' or 'P accurately describes reality.' The alternatives to the correspondence theory only proliferate confusion and miscommunication.

Given that a true theory is one which corresponds to the facts, there are some kind of sentences which cannot be true. For example, the setence 'every x is y' cannot be true, because the letters x and y are variables i.e. there can be no correspondence to the facts because there is nothing for the facts to correspond to. The letters must be interpreted before the sentence can correspond to the facts, as in the sentence 'every raven is black'. Another example of a sentence which cannot be true is any sentence featuring a pronoun, such as 'he is a black raven'. The pronoun 'he' is a variable, and without any context in which it can be interpreted the sentence cannot be true.

In life the decisions we make depend upon what we think is true: we organise our activities around those beliefs in the hope of achieving our desired ends. One thing which we cannot do is act as though a sentence with variables in it is true, because such a sentence cannot correspond to the facts. Here arises a problem for anyone who believes that our decisions depend upon what we think is probably true, because sentences asserting probabilities always include variables, cannot correspond to the facts and, therefore, cannot be true.

Let the probability of rolling a 'five' with a die be 1/6, and imagine the following sequence of rolls:

four, two, six, four, three, one, five, one, two, two, ... ad infinitum

My contention here is the sentence 'the probability of rolling a "five" is 1/6' can be more accurately written 'the probability of roll x being a "five" is 1/6', or in other words, the probability of obtaining a 'five' by random selection from this sequence is 1/6.

However, the sentence 'the probability of roll x being a "five" is 1/6' contains a variable, and therefore, cannot correspond to the facts. In conseuquence, it is not possible to make decisions based on any sentence asserting probability -- no sentence which can correspond to the facts can be probable, since no sentence asserting a probability can correspond to the facts. The phrase 'probably true' does not make sense. The variable x must be interpreted before the sentence can be intelligibly said to be true or false. However, such an interpretation would result in a sentence like 'the probability of the sixth roll being a "five" is 1/6', which is false. There was only one sixth role, and for a sequence with one member the probability that it is a 'five' is either 1 or 0 i.e. true or false.

Hopefully that is a little clearer.
 

FDG

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That's a philosophical objection I had in mind since I first came into contact with probability. It's actually one of the best lame excuses to spend a lot of time playing the lottery and the like: after all, if you actually win, the probability of winning is one.
 

redacted

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However, the sentence 'the probability of roll x being a "five" is 1/6' contains a variable, and therefore, cannot correspond to the facts. In conseuquence, it is not possible to make decisions based on any sentence asserting probability -- no sentence which can correspond to the facts can be probable, since no sentence asserting a probability can correspond to the facts. The phrase 'probably true' does not make sense.

i get your point but who cares about truth if you define it that way? people are using a differently defined 'truth' when they say "probably true". they're not wrong. "x is probably true" really means something like "i have a model in my head of how i believe the world works with some functions relating objects in the model to each other. if i apply all of those functions to the current state of the model, the next state is x. i am assuming that this model will correspond to the world."

inference is useful. if you want to change some words around in the definition so that it doesn't say anything about truth, that's fine. i have a feeling that's as far as you'd want to go anyway.

and what exactly are "facts"?

also, what if someone said "x is probably going to be true" instead of "x is probably true"?
 

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Does this thread remind anyone else of Gary Oldman constantly flipping a coin (and landing "heads") in "Rosencrantz and Guildenstern Are Dead"?
 
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Does this thread remind anyone else of Gary Oldman constantly flipping a coin (and landing "heads") in "Rosencrantz and Guildenstern Are Dead"?

Sort of. In the sense that I'd rather be looking at anything else and yet for some reason I won't.
 

Bluesman

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May I recommend that the original poster do some research into Bayesian vs. Frequentist schools of thought in statistics. It might address some of your thoughts.
 

reason

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May I recommend that the original poster do some research into Bayesian vs. Frequentist schools of thought in statistics. It might address some of your thoughts.
You may, though you needn't. I am aware of the Bayesian vs. Frequentist thing, though do not quite understand of why there is a 'vs' in the middle of the two.

Anyway, even my recent restatement is flawed, in several ways. I am still working on these ideas, and they just keep getting more and more interesting. At this moment in time I am primarily concerned with "frequentism", but plan to come to Bayesianism later.
 

Udog

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If you strip away all the fanciness, Monty Hall boils down to the fact that the host eliminates one of the wrong answers. His guess is not random.

So, what started out for you as a 1 in 3 guess becomes a 1 in 2 guess, if you decide to switch doors. If you do not switch doors, your guess still remains in its original 1 in 3 flavor, despite the host removing a door.

Hah, don't know if that makes it any clearer!
 
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ygolo

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My contention here is the sentence 'the probability of rolling a "five" is 1/6' can be more accurately written 'the probability of roll x being a "five" is 1/6', or in other words, the probability of obtaining a 'five' by random selection from this sequence is 1/6.

However, the sentence 'the probability of roll x being a "five" is 1/6' contains a variable, and therefore, cannot correspond to the facts. In conseuquence, it is not possible to make decisions based on any sentence asserting probability -- no sentence which can correspond to the facts can be probable, since no sentence asserting a probability can correspond to the facts. The phrase 'probably true' does not make sense. The variable x must be interpreted before the sentence can be intelligibly said to be true or false. However, such an interpretation would result in a sentence like 'the probability of the sixth roll being a "five" is 1/6', which is false. There was only one sixth role, and for a sequence with one member the probability that it is a 'five' is either 1 or 0 i.e. true or false.

Hopefully that is a little clearer.

Anyway, even my recent restatement is flawed, in several ways. I am still working on these ideas, and they just keep getting more and more interesting. At this moment in time I am primarily concerned with "frequentism", but plan to come to Bayesianism later.

Perhaps, I can rephraze what I believe you are trying to say, and we can see if we are in agreement.

In a frequentist intepretation of probability, the "probability" of a particular outcome of an event is defined to be its long-run reltive frequency. That is the percentage of an infinite number of runs of identical but independent events that have the outcome in question.

However, at any given time, only a finite number of events will have transpired (and in reality these events are not identical and indepedent). That means, the the long-run relative frequency remains nothing more than a guess.

Probability, however is a profoundly useful concept. One that is absolutely required in the science and engineering discuplines.

May I recommend that the original poster do some research into Bayesian vs. Frequentist schools of thought in statistics. It might address some of your thoughts.
This: Monty Hall problem - Wikipedia, the free encyclopedia is so completely and utterly confounding to me, that I am humbled by my intellectual incapacities.

We've covered similar issues earlier (shameless self-promotion):
Simple Puzzles to Stump People
http://www.typologycentral.com/foru...-accident-grrr-frequentists-vs-bayesians.html
 
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