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> doesn't mean that in the long run you will obtain the same number of heads and tails IANAM, but in my understanding that's exactly what it means, if you expr

by paol 10y ago

> doesn't mean that in the long run you will obtain the same number of heads and tails IANAM, but in my understanding that's exactly what it means, if you express "in the long run" as "the ratio tends to 0.5 as the number of tosses tends toward ∞"

danbruc 10y ago
IANAM either, but that seems to be wrong to me. For any finite number of tosses you may certainly not converge to 0.5 even if it becomes very unlikely pretty quickly not to do so. In order for this to be true, it would have to be literally impossible to always get heads on every toss no matter how often you try. But it seems counter-intuitive at the very least that it is inevitable to get tails eventually. What rules out the possibility to get heads over and over again, an infinite number of times? [1] But if it is possible to obtain an infinite number of heads and not a single tails, then it is not true that ratio always converges towards 0.5. In consequence one would have to modify the definition and say that it is only very likely to converge towards 0.5 but that obviously raises the problem that »very likely« sounds a lot like »with high probability« which is the thing we are trying to define, hence this would become a somewhat circular definition. [1] Actually it must not only be impossible to only ever get heads or tails, it must be impossible to obtain any sequence of outcomes that has not exactly the same number of heads and tails. There is only one sequence of outcomes with only heads and only one with only tails but there is already an infinite number of sequences with all heads but one tails or vice versa. And there is an enormous number of possible sequences with 40 % heads and 60 % tails or any other ratio other than 50/50.
- 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.
- daemonomania 10y ago
  If a coin were flipped arbitrarily many times, and it landed on heads each time, and if we had no other information about the coin, a frequentist would say that the probability the coin lands on heads is one. Now, you might protest and say, "But this is a fair coin that just happened to land on heads arbitrarily many times". But then you are relying on a prior definition of probability in order to justify your objection to the frequentist's claim, since presumably what you mean by "fair" is that the probability the coin lands on heads is 1/2.
  - danbruc 10y ago
    I am not sure if that really addresses my issue. My problem is that I don't see what eventually forces convergences to 50/50. We can use two or even better many coins or, at least superficially equivalent, one coin and let several experimenters take turns. The results will converge towards 50/50 but only with high probability, or at least I don't see why they alway would, i.e. why not at least one sequence could not converge indefinitely. And that seems to lead to an infinite tower of experiments. When tossing a coin infinitely often the ratio of heads and tails will converge to 50/50 but only in that sense that you have to toss the coin an infinite number of times an infinite number of times. And you can't really stop here because you may still deviate arbitrarily far from the expected outcome. So you toss the coin an infinite number of times an infinite number of times an infinite number of times. And you have to keep nesting your experiments until you made an infinite tower of experiments. And I have no idea if that would be sufficient to ensure that you can no longer deviate from the expected outcome, my intuition tends towards no, but then again this is certainly far outside of the realm of things for which I would intuition expect to work. Likely I am really just confused and missing something obvious.
    - mvf0 10y ago
      Your problem is that you don't seem to have sufficient understanding of the concept of limits.

by paol 10y ago

danbruc 10y ago

IANAM either, but that seems to be wrong to me. For any finite number of tosses you may certainly not converge to 0.5 even if it becomes very unlikely pretty quickly not to do so. In order for this to be true, it would have to be literally impossible to always get heads on every toss no matter how often you try. But it seems counter-intuitive at the very least that it is inevitable to get tails eventually. What rules out the possibility to get heads over and over again, an infinite number of times? [1] But if it is possible to obtain an infinite number of heads and not a single tails, then it is not true that ratio always converges towards 0.5. In consequence one would have to modify the definition and say that it is only very likely to converge towards 0.5 but that obviously raises the problem that »very likely« sounds a lot like »with high probability« which is the thing we are trying to define, hence this would become a somewhat circular definition. [1] Actually it must not only be impossible to only ever get heads or tails, it must be impossible to obtain any sequence of outcomes that has not exactly the same number of heads and tails. There is only one sequence of outcomes with only heads and only one with only tails but there is already an infinite number of sequences with all heads but one tails or vice versa. And there is an enormous number of possible sequences with 40 % heads and 60 % tails or any other ratio other than 50/50.

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.
daemonomania 10y ago
If a coin were flipped arbitrarily many times, and it landed on heads each time, and if we had no other information about the coin, a frequentist would say that the probability the coin lands on heads is one. Now, you might protest and say, "But this is a fair coin that just happened to land on heads arbitrarily many times". But then you are relying on a prior definition of probability in order to justify your objection to the frequentist's claim, since presumably what you mean by "fair" is that the probability the coin lands on heads is 1/2.
- danbruc 10y ago
  I am not sure if that really addresses my issue. My problem is that I don't see what eventually forces convergences to 50/50. We can use two or even better many coins or, at least superficially equivalent, one coin and let several experimenters take turns. The results will converge towards 50/50 but only with high probability, or at least I don't see why they alway would, i.e. why not at least one sequence could not converge indefinitely. And that seems to lead to an infinite tower of experiments. When tossing a coin infinitely often the ratio of heads and tails will converge to 50/50 but only in that sense that you have to toss the coin an infinite number of times an infinite number of times. And you can't really stop here because you may still deviate arbitrarily far from the expected outcome. So you toss the coin an infinite number of times an infinite number of times an infinite number of times. And you have to keep nesting your experiments until you made an infinite tower of experiments. And I have no idea if that would be sufficient to ensure that you can no longer deviate from the expected outcome, my intuition tends towards no, but then again this is certainly far outside of the realm of things for which I would intuition expect to work. Likely I am really just confused and missing something obvious.
  - mvf0 10y ago
    Your problem is that you don't seem to have sufficient understanding of the concept of limits.