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The probability of a probability?

  1. Aug 8, 2016 #1
    TEXT :
    - We have an event called X with two possible outcomes that are outcome A and outcome B .
    the probability for the outcome to be A is noted as p , where p is a real number in the range [0.0 ; 1.0].
    we also note p' a real number representing the probability that p is in range [0.9 ; 1.0] .
    - additional information "#" : the event X happened a billion times among which the outcome is always A.

    QUESTIONS:

    1)- without noting the additional information "#" , can a value be assigned to p' ?
    (the student has to indicate the value and prove the existence of the value , if a value cannot be
    assigned the student has to prove the non-existence of the value)

    2)-
    a)- if we note the additional information "#", does that change the previous value in anyway ?
    b)- can we assign a new value to p' ?



     
  2. jcsd
  3. Aug 8, 2016 #2

    mfb

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    What did you figure out so far?
    Can you think of example systems that show this type of probability?
    Have you heard of Bayesian statistics?
     
  4. Aug 8, 2016 #3
    thank you for the response sir .
    well the answer to question one is 1/10 it's not that hard .
    i am though unable to solve question 2 in anyway and i am unable to find a proper system to shape this problem . i know the bayes theorem we had that in probability , the probability of something given an other contributing factor , but i am unable to get that to serve here .

    thank you .
     
  5. Aug 8, 2016 #4

    Stephen Tashi

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    This is either a very sophisticated problem involving concepts or a mis-statement of a problem involving Bayes' theorem that intends for the student to do some computations. As stated, the problem doesn't give any distribution for p. It only states the range of possible values of p. If the textbook author wants to pose a computation problem then he needs to give a distribution for p over its range - for example, he could say that p has a uniform distribution on [0,1].



    That is a very sophisticated question - even if the distribution of p is given !

    Given the distribution of p, one could compute the expected value of p', but can we call this expected value the unique value of p' ?

    The sophisticated nature of the question may be unintentional. If this is a problem in an elementary textbook then I think the author would have said that p has a uniform distribution on [0,1] and want the student to compute the expected value of p' and declare it to be "the" value of p'.

    To discuss the question in a sophisticated manner would be interesting because it involves the fundamentals of probability theory - sample spaces and sample spaces underlying conditional probabilities etc.

     
  6. Aug 8, 2016 #5
    thank you sir ,
    the problem is pointed at college students so i think the solution can be relatively complicated .
    i wrote all of the information of the problem and there is no information about the distribution .
    the most unusual thing though is that the book contains solutions for the problems , but that one in specific has no solution within the book .
     
  7. Aug 8, 2016 #6

    Stephen Tashi

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    Is 1/10 the answer given by the textbook ?

    If so then I think the author intends the wording of the problem to imply that p is uniformly distributed on [0,1].

    This thread is in the pre-calculus section, but is the problem from a college textbook that assumes the student knows how to do integration?
     
  8. Aug 8, 2016 #7
    the book is written for college students , and i was quite unable to understand most of what it contains , but this problem in specific seemed interesting to me , but there was no solution for it in the book . that's why i posted it here .
    the '1/10' value is mine , and i think it's probably false .
     
  9. Aug 8, 2016 #8
    Usually the probability of an outcome is determined in a ... deterministic way, so we have that the (Probability of (probability of outcome A happening=p ))= 1 for some p in (0,1) and (Probability of (probability of outcome A happening=x ))= 0 for every other x in (0,1) except the p.

    Anyway if we assume that p is not determined in a deterministic way and thus it has a distribution , and it is a uniform distribution of p, then for 1) it will be p'=1/10.

    for 2) if we take into account that info, and from the definition of "a posteriori probability" we will have that the a posteriori probability of outcome A is very close to 1. Thus the new value for p' will also be close to 1.
     
  10. Aug 8, 2016 #9

    Ray Vickson

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    Using the Bayesian approach in this problem with a so-called uniform prior for ##p## gives an initial value of ##p'= 0.1##, as you suggest. However, the posterior value of ##p'##, given the experimental outcomes you describe, is quit a bit different: I get
    $$ \text{posterior value of }\, p' = 1-0.2474955439 \times 10^{-45757490} $$
    This is pretty close to 1: it is 0.999.... with about 45,000,000 '9's after the decimal point.

    You need calculus to get this, and a having access to results from a more advanced probability course would be helpful as well.
     
    Last edited: Aug 8, 2016
  11. Aug 8, 2016 #10

    Stephen Tashi

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    Is it a mathematics textbook about probability ? - or a collection of varied mathematical problems? - or a collection of problems from all sorts of disciplines ?

    I agree that the problem is interesting. I think elementary textbooks deliberately avoid posing problems that explicitly involve "the probability of a probability".

    If we assume p is uniformly distributed on [0,1] then I agree with your answer.

    As to question 2), are you interested in a qualitative answer or in detailed computations?
     
  12. Aug 8, 2016 #11

    mfb

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    The answer to problem 2 is easy: it is the same for all reasonable priors. The answer to problem 1 is more interesting.
    There is no reason to assume this, however.

    Take a coin, the two sides are the outcomes A and B. What is your prior for its distribution of p?
     
  13. Aug 8, 2016 #12

    haruspex

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    Sounds to me as though @alan_longor thinks that is obviously the answer, but the textbook does not provide one.
    If so, I suggest that no prior should be assumed and that the point of the first part is to prove that no value can be assigned.
     
  14. Aug 8, 2016 #13
    my problem is the fact that this book seems to contain some absurd things that i have never heard of , here is an other problem i came across that is supposed to follow that one . this one got me confused even more .


    TEXT :

    we have an event X that has two possible outcomes A and B . for the purpose of this problem we will set a definition , a "special" event means to us now an event that we can set a probability for it's outcome , which means that if X is a special event , there exists a real number p where p would be in range [0.0 ; 1.0] and where p represents the probability for the outcome of X to be A , if the event is not special we cannot assign a fixed real number as a probability for it's outcome .

    > the following is only meant for demonstration purposes :
    > let the event X happen N times , where n is the number of times that A is the outcome .
    > if X is special then if N -> oo then n/N -> p , where p is a real fixed number in range [0.0 ; 1.0]
    > if X is not special then if N -> oo then n/N is indefinite but always remains in range [0.0 ; 1.0]


    now let p' be a real number in range [0.0 ; 1.0] that describes the probability that event X is special , which is the probability that p exists .

    additional information # : event X happened a billion times among which the outcome is always A .

    QUESTIONS :

    1)- without noting the additional information "#" , can a value be assigned to p' ?
    (the student has to indicate the value and prove the existence of the value , if a value cannot be
    assigned the student has to prove the non-existence of the value)

    2)-
    a)- if we note the additional information "#", does that change the previous value in anyway (if it exists) ?
    b)- can we assign a new value to p' (if it exists) ?

    3)- if we note the additional information "#" , let p" be a real number in range [0,0 ; 1.0] where p" would represents the probability that the outcome of the next attempt is B ( attempt number 10^(9) + 1 ) (if p" exists)
    a)- can p" exist for any value of p' (if p' exists) ?
    b)- can we assign a value to p" ?
     
    Last edited: Aug 8, 2016
  15. Aug 8, 2016 #14

    haruspex

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    It's tough to get your head around, but it does not strike me as absurd. Not sure that I can explain it any more clearly, but if you care to indicate where you get lost and why then I'll try.
     
  16. Aug 8, 2016 #15

    Stephen Tashi

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    This doesn't sound like material from routine probability text! What is the title of this book?

    The only interpretation of that text than I can make in terms of probability theory is that a "special" event is an event with a constant probability that can be subjected to an independent set of trials. An non-special event could be interpreted as an "event" in the sense of common speech - for example, "The traffic light at the intersection near my house was red when I got to it". We can conduct a set of trials of a non-special event, but the result doesn't have the same probability of occurrence on each of the trials.


    Those statements illustrate the difference between "probability" and "observed frequency".

    Another way to put the same question is "Given an event X, let E be the event that X is a special event. Is E a special event or a non-special event?"
     
  17. Aug 8, 2016 #16

    haruspex

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    I don't think that would be sufficiently badly behaved. Over arbitrarily long trials, the frequency would still converge. The events are, at base, controlled by atomic events with fixed probabilities.
    We can take the definition of nonspecial as being that the probability does not converge. Imagine e.g. a demon who is tracking the outcomes and adjusting the probability. If the frequency up to some point is f, the demon could then set the probability to a value f' that differs from f by 0.5. As the trials proceed, the frequency (over the entire sequence) approaches f', whereupon the demon adjusts it again.
     
  18. Aug 8, 2016 #17

    Stephen Tashi

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    I'll disagree. We don't know that an arbitrarily chosen event is controlled by atomic events with fixed probabilities.

    Defining a "repeatable event" (or repeatable experiment) is essentially a contradiction in terms because if we exactly repeated something, it would not be a "different" event or experiment. So when we define a "repeatable" event we specify a set of different things that will be regarded as "the same event". This necessary ambiguity in the definition of an repeatable event is sufficient to produce a situation where the probability of an event on a given trial is not constant. The probability that a result occurs on a given trial is distinct from the probability that the result occurs on a randomly selected trial.


    That's a good idea, but it only pushes the requirement of being non-special onto the demon. Can the demon be implemented by a deterministic algorithm?

    If there was a demon, "Even Steven", that implemented the commonly held fallacy that a strings of heads in tosses of a coin should made a tail more likely then Even Steven could produce a frequency of heads near 0.5, but the event "head" wouldn't have the same probability on each trial. If we pick one of the tosses of Even Steven at random then the probability that the result is heads is about 0.5.

    (I can see why the book didn't try to answer to this probem!)
     
  19. Aug 9, 2016 #18

    haruspex

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    Yes, I specified it, pretty much. We can make it simpler. The demon fixes the outcomes as 01100001111111100000000000000.... Clearly the "average over the first n" gives a sequence that does not converge.
    This is a somewhat different issue, that of probability given what? The initial, description of special event doesn't specify whether the history of preceding outcomes is known. If it is unknown then Even Steven does achieve an evens chance on any given trial. But that description is not really a usable definition. The convergence definition provided later says that Even Steven is 'special'. In fact he would be extra special, converging more rapidly than independent coin tosses.
     
  20. Aug 9, 2016 #19

    Stephen Tashi

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    I see what you mean.

    I think it's a convergence theorem instead of a definition. It's also a garbled theorem because it omits reference to probability. It guarantees that n/N "approaches" p as N approaches infinity. It doesn't define what "approaches" means in that context. It isn't a statement of "The Law Of Large Numbers".
     
  21. Aug 9, 2016 #20

    haruspex

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    Yes, the notion of convergence in the context of random variables needs to be defined first.
     
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