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Thread: On Probability

  1. #1
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    Default On Probability

    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
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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.

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    Quote Originally Posted by Jack Flak View Post
    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).
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    Quote Originally Posted by reason View Post
    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.

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    Quote Originally Posted by Jack Flak View Post
    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.
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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.

  7. #7

    Default 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.

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    Quote Originally Posted by reason View Post
    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.
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    Quote Originally Posted by IlyaK1986 View Post
    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).
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    Quote Originally Posted by ygolo View Post
    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.
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