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B How might determinism affect probability?

  1. Jun 28, 2018 #1
    The probability, I was taught, for red to appear at roulette (on a European table, with a single green zero) is 18/37; and the probability for non-red to appear (black or green zero) is 19/37. If we live in a deterministic universe (I appreciate that that’s a big if for some, not so much for others) is the actual probabilites for any given spin one and zero (1 and zero) e.g, if it’s been determined that red 36 will appear then it’s 100% guaranteed that red 36 will appear (and 100% guaranteed that non-red will appear. So, in that situation, is the probability for red 1, and the probability of non-red 0?

    P.s. I appreciate that, unless you were Laplace’s demon, you couldn’t know this in advance of the spin but I’m still wonderin if, even though we would describe the probabilites as 18/37 and 19/37, they would actually be 1 and 0.
     
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  3. Jun 28, 2018 #2

    Math_QED

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    This is rather a philosophical question, and it doesn't make sense to me to talk about probability in a deterministic context.
     
  4. Jun 28, 2018 #3

    Dale

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    There are at least two different schools of thought regarding probabilities.

    One is the “frequentists” which view probability similarly to how you are suggesting. Probabilities only exist as the fraction of results with a given outcome after an infinite number of trials. So probability is not defined for a single spin. You would have to spin an infinite number of times and get red each time for the probability to be 1.

    The other is the “Bayesians” which take probability as a measure of our belief. For them probability of a single event is well defined, and can even be measured in terms of hypothetical bets. Even if the universe is truly deterministic, we may not know enough to determine the outcome, so the probability would account for our ignorance.
     
  5. Jun 29, 2018 #4
    I'm not sure if the determinism part is only philosophical. I've spoken to scientists who see determinism as the 'only game in town' due to their belief in causation (although some scientists I've spoken to believe quantum indeterminacy means we can't be certain of determinism). So for some it seems to be a physics question as well as a philosophical one. It seems to me to be a maths question also. If I asked what is the probability of non-red appearing on a wheel with no non-red pockets my guess is that most mathematicians would say it's zero. The above roulette wheel (in the universe where it's determined that red will appear) is analogous to the wheel with no non-red pockets, which is why I'm wondering if the probability is also zero.
     
  6. Jun 29, 2018 #5
    “One is the “frequentists”...”

    Which of these two schools of thought dominate maths (I’m thinking of your average text book which tells us that the P of rolling a 6 on a die is 1/6 etc)?

    “...after an infinite number of trials.”

    Is this the idea that we will see exactly 18/37 reds and 19/37 non-reds over an infinite number of trials? Or exactly 50% heads, 50% tails etc?

    “So probability is not defined for a single spin.”

    So it’s not the frequentists who are writing your average text book?

    “You would have to spin an infinite number of times and get red each time for the probability to be 1.”

    Which can’t happen, even in theory, given that there’s 18 red pockets, 19 non-red pockets, yeah?

    “For them (the Bayesians) probability of a single event is well defined, and can even be measured in terms of hypothetical bets.”

    What would a Bayesian say about a single spin of the wheel regards red v non-red?
     
  7. Jun 29, 2018 #6

    BWV

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    There is no question that the physics of a roulette ball are completely deterministic - if you know precisely all the relevant variables you can predict the outcome with 100% accuracy. But, because the outcome is extremely sensitive to initial conditions - the force the ball is dropped, the spin rate of the wheel, where the ball lands on the wheel etc, it is effectively random and the use of probability reflects our ignorance of the outcome - i.e. in the absence of any relevant information, any particular spot on the wheel is as likely as any other.
     
  8. Jun 29, 2018 #7

    Stephen Tashi

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    We must distinguish between mathematics and the application of mathematics. The mathematics of probability theory makes no comment of how physical situations produce probabilities. It doesn't specify any relation between probabilities and the actual frequencies with which events occur. If a math textbook states something about the probability of rolling a 6 with a "fair" die, it is merely a hypothesis in a problem - in the same sense that a statement like "Ed has 6 apples" can be a hypothesis (a "given") in algebra problem. As far as the pure mathematics of probability theory goes, it is a special case of "measure theory". There is no distinct "frequentist measure theory" versus "Bayesian measure theory".

    It is in applications of probability theory that there is a distinction between the Bayesian and frequentist approaches. To get mathematical solutions to real life problems, assumptions must be made. The frequentist and Bayesian approaches differ in how they choose to make assumptions. When it comes to applying probability theory to physics, there are many different interpretations of probability. In fact, there are many different interpretations of probability among people who call themselves Bayesians.

    Textbooks on probability theory often try to be helpful by mixing the mathematics of probability with examples of its application. For example, the typical text on introductory statistics teaches the "frequentist" approach to applying probability theory. Modern textbooks about probability theory (as opposed being only about the topic of statistics) usually show both the frequentist and Bayesian approaches.
     
  9. Jun 29, 2018 #8

    Stephen Tashi

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    Of course, you mean the Newtonian physics.
     
  10. Jun 29, 2018 #9

    Mark44

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    I'm thinking that maybe you don't understand probability.

    This isn't a term that mathematicians use to describe themselves, to the best of my knowledge.

    Are you questioning whether this probability is correct?
     
  11. Jun 29, 2018 #10

    Dale

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    The frequentist school of thought is by far more common. Although both will give you the same p for rolling a fair die.

    Yes, assuming tautologically that it is a fair wheel or a fair coin.

    It is the frequentists, but the question you asked is not something that is addressed in standard courses.

    That depends on what prior information they have. If they know with complete certainty the determined outcome then they would say 1 or 0 accordingly. Otherwise they would take any limited information they have into account.
     
    Last edited: Jun 29, 2018
  12. Jul 2, 2018 #11

    “...the use of probability reflects our ignorance of the outcome - i.e. in the absence of any relevant information, any particular spot on the wheel is as likely as any other.”

    I appreciate that assuming it is deterministic (as you say it is) we are still ignorant of what the outcome will be; but I don’t see how “any particular spot on the wheel is as likely as any other” if it is deterministic. I can see why from our subjective point of view it would seem like that, given our ignorance of the outcome, but the objective reality would be that THERE IS ONLY ONE POSSIBLE OUTCOME, i.e, the ball will land in a red pocket because that is what has been determined by antecedent events. Which is why I’m wondering if the actual probability for red is 1, and the actual probability for non-red is 0.
     
  13. Jul 2, 2018 #12

    Thanks, Stephen. I looked up measure theory but it’s too advanced for me at the moment. Can you tell me in the meantime if I’ve understood part of what you said correctly. Pure maths, regards probability, is not based on any number of observed events (e.g, looking at thousands of spins of a roulette wheel) it’s purely theoretical. We ASSUME that with an unbiased roulette wheel and an unbiased croupier the ball has an equal chance of landing in any of the pockets and assign a probability accordingly (the P for each pocket is 1/37)?

    “The frequentist and Bayesian approaches differ in how they choose to make assumptions.”

    Could you give me some examples of how they might differ in their assumptions regards my example of red v non-red?
     
  14. Jul 2, 2018 #13

    Dale

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    In the Bayesian approach probability does not reflect objective reality. In the frequentist approach it represents the objective reality of the proportion in an infinite ensemble of experiments. So a frequentist cannot assign a probability for a single event.
     
    Last edited: Jul 2, 2018
  15. Jul 2, 2018 #14

    BWV

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    If what you say was true, then (outside of quantum mechanics) the odds for anything would be either zero or one and statistics would be of absolutely no use to anyone If I flip a coin, look at the outcome and ask you to guess heads or tails, then, yes, the probability depends on your ignorance. Just accept probability as a measure of ignorance about a system and stop overthinking it, like it represents some objective reality.

    And ignore the bayesian / frequentist stuff - its a huge waste of time going down that rabbit hole.
     
  16. Jul 2, 2018 #15

    Dale

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    I disagree completely. The source of his confusion is trying to put a frequentist interpretation of probability where it cannot be applied. The OP needs to be aware that there is an alternative interpretation where it can be applied but in that interpretation probability is not objective.

    Which is the distinction between the frequentist and Bayesian interpretations of probability, in a nutshell.
     
  17. Jul 2, 2018 #16

    BWV

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    The Bayesian / Frequentist stuff does not apply to the OP, is is confused on a more basic level - confusing assumptions of probability with basic physics which are separate issues. If you cant apply frequentist definitions of probability to casino games, then where can you use it? (of course you can, this is where the whole theory developed) Its is not a B/F issue to state that probability involves our ignorance of certain attributes of a system, as outside of QM, this is where all probability comes from.
     
  18. Jul 2, 2018 #17

    Stephen Tashi

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    One way to conceptualize measure theory is that it is an abstract treatment of the idea of "area" or "volume". A set (such as the possible outcomes of the toss of a fair coin, {H,T}) has certain subsets (such as {H},{T}) for which there is a function p() that assigns a "measure" to the subsets (such as p{H} = 1/2, p{T} = 1/2). No interpretation of what the measure means is given.

    Yes. The pure mathematical treatment of probability does not specify any physical experiments for measuring probabilities.

    I'll say no.

    Making that statement seems to assume that "unbiased" is concept from physics and that "chance of" has either a physical definition or a mathematical definition that is more fundamental than the phrase "probability of". If you define "unbiased roulette wheel" to mean a wheel where each outcome has an equal "chance of" happening and define "chance of" to mean "probability of" then you can make the above statement - however, the statement is then "true by definition"; it has no significant content other than being a vocabulary exercise.

    We have to face the fact that a logically consistent concept of "probability" contradicts any attempts to make definite assertions that do not themselves involve the concept of "probability". Probability theory is circular in that respect. For example if the probability that a "fair" coin is tossed 100 times, we can say nothing definite about whether it lands heads 50 times or whether it lands heads between 40 and 60 times. We can only compute the probability that it lands heads 50 times and the probability it lands heads between 40 and 60 times. Probability theory only makes statements about probabilities.

    The circular nature of probability theory is unsatisfactory from the point of view of confronting real life problems. However the circular nature of probability theory is essential from the point of view of doing mathematics. Attempts to make probability theory say something definite about things other than probabilities lead to metaphysical disputes. (e.g. Can an event with probability zero happen? If I toss a fair coin long enough, will I (definitely) eventually get 3 times as many heads as tails? etc.) Mathematical probability theory wisely does not touch such questions!

    Let say you have a roulette wheel not know to be "fair".

    A frequentist approach is to assume the probability of red is a some fixed, but unknown number ##P_{red}##. We assume the wheel is fair, i.e. we assume ##P_{red} = 18/37##. We compute the probability ##p## that in 3700 results from the wheel we would get between ##1800 - T## and ##1800 + T## reds. We choose ##T## so that ##p## is "small" (e.g. p = 0.05 is a typical choice). We observe 3700 outcomes of the wheel. If the number of reds is less than ##1800-T## or greater than ##1800 + T## we "reject" the hypothesis that the wheel is fair.

    In the frequentist approach, the choice of ##T## is subjective. The statement that we "reject" the hypothesis is not the same as saying definitely "The wheel is not fair" or asserting that the probability the wheel is not fair is such-and-such. The frequentist approach is simply a procedure which has been found empirically to be useful in many fields of study.

    A Bayesian approach would be to imagine the wheel in question was taken at random from a population of wheels and this results in some probability distribution for the number ##P_{red}##. (In this example, you must swallow the notion of "a probability of a probability"). For example we might assume the ##P_{red}## is chosen from a uniform distribution of numbers over the interval [0.4, 0.7]. We observe 3700 outcomes of the wheel and observe ##N_{red}## reds. Then we compute the function that gives us the probability that ##P_{red} = x ## given there were ##N_{red}## reds. (Imagine making a graph of this function.) Making a decision about whether the wheel is fair or not is done on the basis of that function and other assumed or known information that quantifies the expected gain or loss from making a correct or incorrect decision.

    A fundamental distinction between the frequentist and Bayesian approaches is that the frequentist procedure involves computing the probability of the data given the hypothesis that the wheel is fair while the Bayesian procedure involves computing the probability that ##P_{red}## takes various values given the observed data. (There is a distinction between the conditional probabilities Pr(A given B) and Pr(B given A) ).

    The frequentist and Bayesian procedures also differ in what subjective assumptions must be made. The Bayesian approach requires more elaborate assumptions.

    People can offer various philosophical reasons for using the above approaches (e.g. that ##P_{red}## is an objective property of the physical wheel versus that ##P_{red}## is quantification of our ignorance about whether a red happens etc.). The mathematical procedures themselves do not specify particular philosophies or metaphysics. However, it's fair to say that metaphysical views about probabilities being a measure of information or ignorance can be used to justify Bayesian procedures.
     
  19. Jul 2, 2018 #18

    BWV

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    A simpler Bayesian approach would be to say, that in the absence of other information, how the wheel is 'unfair' is unknown, so the prior for red would simply be
    18/37, as the 'fair' odds would be the mean of a distribution of 'unfair' wheels. As the wheel is spun an arbitrary number of times and the 'unfairness' begins to emerge, then the Bayesian would update the distribution accordingly.
     
  20. Jul 2, 2018 #19

    Stephen Tashi

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    A Bayesian guess would be 18/37, but for a prior that can be updated, we'd need a distribution where other values are possible. (i.e. the prior distribution ##Pr(P_{red} = 18/37) = 1## can't be updated .)
     
  21. Jul 2, 2018 #20

    BWV

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    No, the P(red)=18/37 is the mean of the prior distribution of 'unfair' wheels
     
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