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> For any finite number of tosses you can certainly not converge to 0.5 I think you mean "may not" rather than "can not". But yes, that's why you need to resor

by paol 10y ago

> For any finite number of tosses you can certainly not converge to 0.5 I think you mean "may not" rather than "can not". But yes, that's why you need to resort to limits[0] to understand the problem. [0] https://en.wikipedia.org/wiki/Limit_of_a_sequence https://en.wikipedia.org/wiki/Limit_of_a_sequence

danbruc 10y ago
Thanks, fixed. It understand that it is the limit, but the question I am asking is whether it is possible to always get heads, an infinite number of times. If this is possible, then it is not true that the relative frequency always approaches 0.5 as the number of repetitions goes to infinity. I understand that the probability of obtaining heads an infinite number of times is zero, but does that mean it is impossible? After all any specific infinite sequence of outcomes has probability zero.
- paol 10y ago
  > the question I am asking is whether it is possible to always get heads, an infinite number of times No. By my definition (lay, may be wrong), the probability is the ratio of heads to tails to which we converge at the limit (infinity). The only way for it to converge to 0 is if the probability of heads was 0 to begin with, which would contradict the initial assertion that it was a fair coin. By the way, from your previous comment it seems to me that you're struggling with getting an intuitive grasp on arguments that rely on infinity. To most people (me included for sure) it's initially difficult to accept that infinity is qualitatively different than a finite amount. Our brain seems to superficially accept the concept, but really it keeps trying to reason about it as a really large (but still finite) quantity. Are you familiar with the argument of whether 0.999... equals 1? (Spoiler: it does) Some people have a really hard time coming to terms with it, and the fundamental problem for many comes down to the same difficulty of reasoning about infinity. Also, it's a fun way to troll people :)
  - danbruc 10y ago
    No. By my definition (lay, may be wrong), the probability is the ratio of heads to tails to which we converge at the limit (infinity). The only way for it to converge to 0 is if the probability of heads was 0 to begin with, which would contradict the initial assertion that it was a fair coin. This seems problematic to me. You can not perform an infinite number of coin tosses and know to what the relative frequency converges. So how do you conclude that a fair coin has probability of 0.5 for heads then? I can't really put my finger on it, but that argument is somehow circular. The probability is what the relative frequency converges to and I can not get anything other than 0.5 because that means to probability would have to have been not 0.5 to begin with. I can really only say that I disagree, I just can't pin it down exactly. Are you familiar with the argument of whether 0.999... equals 1? (Spoiler: it does) Some people have a really hard time coming to terms with it, and the fundamental problem for many comes down to the same difficulty of reasoning about infinity. I am aware of that and it seems totally obvious to me. And I also certainly know that infinity is not just a really huge number, but I am not sure I internalized that well enough to not make any mistakes because of the difference. Actually I am pretty sure I make mistakes because of that.
- kgwgk 10y ago
  There are different ways to "approach 0.5 as the number of repetitions goes to infinity": https://en.wikipedia.org/wiki/Convergence_of_random_variables https://en.wikipedia.org/wiki/Convergence_of_random_variable... I think in this case you have "convergence in probability" (but I've not read carefully the discussion). There is a stronger form, "almost sure convergence", where there is exact convergence with probability one (i.e. in some cases there is no convergence, but those happen with probability zero).
  - danbruc 10y ago
    That is exactly what I mean. We try to establish what a probability of 0.5 means, i.e. that the relative frequency converges to 0.5 if the number of trails goes to infinity, but that is not exactly true because there is a (possibly vanishingly small) set of outcomes where the relative frequency does not converge to 0.5. This in turn forces us to state that the relative frequency only converges with high probability, but now we have used a probability in our definition of probability, i.e. we made some kind of circular argument. I just skimmed the article you linked and I am not sure if I read it before so I will read it again, but from a first glance it does not look like any definition in there is suitable to avoid the problem.
    - kgwgk 10y ago
      There is no circularity. You're discussing the real-world interpretation of the probability of an event in terms of long-term frequencies. The probability in the definition of convergence is a well-defined mathematical concept that has nothing to do with frequencies. Edit: thinking more about it, I agree that the passage from mathematical probability to real-world probability is not very satisfactory. But it's not really surprising, the very notion of "real-world" is troublesome.
- effie 10y ago
  "whether it is possible to always get heads, an infinite number of times" No if we actually make the experiment, for we cannot toss coin infinitely many times. Yes, if we are just considering and measuring all possible infinite sequences. Measure of such mathematically possible sequence is zero, like measure of a point somewhere in a circle is zero. The point exists and is a possible result of choosing a point, but its probability is zero. Thus zero probability does not mean thing is impossible; it is only very unlikely.
  - danbruc 10y ago
    If an infinite sequence of all heads is a possible outcome, then it is not true that the relative frequency converges to 0.5 in the limit. It is obviously true for many infinite sequences but if all heads is a possibility then it is not true in general. Just to clarify, I am talking about the frequentist's claim that in the long run the relative frequency becomes the probability, i.e. P(x) = lim[n -> ∞] nx / n where n is the number of trails and nx is the number of trails that yielded x. So again, if an infinite sequence of heads is a possible outcome, then that would mean P(heads) = 1 while the frequentist asserts that it necessarily must become 0.5.
    - effie 10y ago
      Does the frequentist view really claim that infinite sequence nx/n always converges, for all possible runs? That is obviously false. The laws of large numbers only state that probability that the limit of nx/n is equal to 1/2 is 1. This is because all sequences are equally probable and there are overwhelmingly many more sequences that limit to 1/2 than sequences that do not. That is different from saying that the limit is always equal to 1/2, because there are these exceptional cases of low measure like HHHHH..., HTTHTTHTT...

by paol 10y ago

danbruc 10y ago

Thanks, fixed. It understand that it is the limit, but the question I am asking is whether it is possible to always get heads, an infinite number of times. If this is possible, then it is not true that the relative frequency always approaches 0.5 as the number of repetitions goes to infinity. I understand that the probability of obtaining heads an infinite number of times is zero, but does that mean it is impossible? After all any specific infinite sequence of outcomes has probability zero.

paol 10y ago
> the question I am asking is whether it is possible to always get heads, an infinite number of times No. By my definition (lay, may be wrong), the probability is the ratio of heads to tails to which we converge at the limit (infinity). The only way for it to converge to 0 is if the probability of heads was 0 to begin with, which would contradict the initial assertion that it was a fair coin. By the way, from your previous comment it seems to me that you're struggling with getting an intuitive grasp on arguments that rely on infinity. To most people (me included for sure) it's initially difficult to accept that infinity is qualitatively different than a finite amount. Our brain seems to superficially accept the concept, but really it keeps trying to reason about it as a really large (but still finite) quantity. Are you familiar with the argument of whether 0.999... equals 1? (Spoiler: it does) Some people have a really hard time coming to terms with it, and the fundamental problem for many comes down to the same difficulty of reasoning about infinity. Also, it's a fun way to troll people :)
- danbruc 10y ago
  No. By my definition (lay, may be wrong), the probability is the ratio of heads to tails to which we converge at the limit (infinity). The only way for it to converge to 0 is if the probability of heads was 0 to begin with, which would contradict the initial assertion that it was a fair coin. This seems problematic to me. You can not perform an infinite number of coin tosses and know to what the relative frequency converges. So how do you conclude that a fair coin has probability of 0.5 for heads then? I can't really put my finger on it, but that argument is somehow circular. The probability is what the relative frequency converges to and I can not get anything other than 0.5 because that means to probability would have to have been not 0.5 to begin with. I can really only say that I disagree, I just can't pin it down exactly. Are you familiar with the argument of whether 0.999... equals 1? (Spoiler: it does) Some people have a really hard time coming to terms with it, and the fundamental problem for many comes down to the same difficulty of reasoning about infinity. I am aware of that and it seems totally obvious to me. And I also certainly know that infinity is not just a really huge number, but I am not sure I internalized that well enough to not make any mistakes because of the difference. Actually I am pretty sure I make mistakes because of that.
kgwgk 10y ago
There are different ways to "approach 0.5 as the number of repetitions goes to infinity": https://en.wikipedia.org/wiki/Convergence_of_random_variables https://en.wikipedia.org/wiki/Convergence_of_random_variable... I think in this case you have "convergence in probability" (but I've not read carefully the discussion). There is a stronger form, "almost sure convergence", where there is exact convergence with probability one (i.e. in some cases there is no convergence, but those happen with probability zero).
- danbruc 10y ago
  That is exactly what I mean. We try to establish what a probability of 0.5 means, i.e. that the relative frequency converges to 0.5 if the number of trails goes to infinity, but that is not exactly true because there is a (possibly vanishingly small) set of outcomes where the relative frequency does not converge to 0.5. This in turn forces us to state that the relative frequency only converges with high probability, but now we have used a probability in our definition of probability, i.e. we made some kind of circular argument. I just skimmed the article you linked and I am not sure if I read it before so I will read it again, but from a first glance it does not look like any definition in there is suitable to avoid the problem.
  - kgwgk 10y ago
    There is no circularity. You're discussing the real-world interpretation of the probability of an event in terms of long-term frequencies. The probability in the definition of convergence is a well-defined mathematical concept that has nothing to do with frequencies. Edit: thinking more about it, I agree that the passage from mathematical probability to real-world probability is not very satisfactory. But it's not really surprising, the very notion of "real-world" is troublesome.
effie 10y ago
"whether it is possible to always get heads, an infinite number of times" No if we actually make the experiment, for we cannot toss coin infinitely many times. Yes, if we are just considering and measuring all possible infinite sequences. Measure of such mathematically possible sequence is zero, like measure of a point somewhere in a circle is zero. The point exists and is a possible result of choosing a point, but its probability is zero. Thus zero probability does not mean thing is impossible; it is only very unlikely.
- danbruc 10y ago
  If an infinite sequence of all heads is a possible outcome, then it is not true that the relative frequency converges to 0.5 in the limit. It is obviously true for many infinite sequences but if all heads is a possibility then it is not true in general. Just to clarify, I am talking about the frequentist's claim that in the long run the relative frequency becomes the probability, i.e. P(x) = lim[n -> ∞] nx / n where n is the number of trails and nx is the number of trails that yielded x. So again, if an infinite sequence of heads is a possible outcome, then that would mean P(heads) = 1 while the frequentist asserts that it necessarily must become 0.5.
  - effie 10y ago
    Does the frequentist view really claim that infinite sequence nx/n always converges, for all possible runs? That is obviously false. The laws of large numbers only state that probability that the limit of nx/n is equal to 1/2 is 1. This is because all sequences are equally probable and there are overwhelmingly many more sequences that limit to 1/2 than sequences that do not. That is different from saying that the limit is always equal to 1/2, because there are these exceptional cases of low measure like HHHHH..., HTTHTTHTT...