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> The question is what difference it makes going from a large but finite to an infinite number of tosses. Convergence is only guaranteed as N -> ∞. The differe

> The question is what difference it makes going from a large but finite to an infinite number of tosses. Convergence is only guaranteed as N -> ∞. The difference between large but finite and infinite is.. well, infinite :) So that's a pretty significant difference. > Either is impossible to get only heads an infinite number of times, then I have a problem understanding why that is, or all heads is still a possible outcome even in the infinite case, then the process does not necessarily yield 0.5 even in the limit of an infinite number of tosses. It's the former. It is impossible to get only heads an infinite number of times. It is possible, but increasingly unlikely, to get only heads a REALLY LARGE, but finite, number of times.

danbruc 10y ago
Okay, assuming that is true, is there an intuitive way to understand that? And the fact that lim[n-> ∞] 0.5^n = 0 unfortunately won't do for me, that is true for any specific infinite sequence, even those containing 50/50 heads and tails. I think most of the sequences - hand-waving, most of an infinite set - are 50/50 heads and tails just because there are more possibilities - hand-waving again - to arrange 50/50 heads and tails versus 100/0 or 40/60 heads and tails. So can I sum over all the 50/50 sequences and get 1 and sum over the rest and get 0? I still would not really understand what forces my coin to show tails eventually, but if there were a measure showing that those two sets have measure 1 respectively 0 it would already be easier to swallow.
- kareemsabri 10y ago
  > Okay, assuming that is true, is there an intuitive way to understand that? And the fact that lim[n-> ∞] 0.5^n = 0 unfortunately won't do for me, that is true for any specific infinite sequence, even those containing 50/50 heads and tails. That's true, but only if you pick ONE specific sequence. The likelihood of any precise sequence of heads/tails is equally unlikely as all heads. However, we are not fixing the outcomes ordering, but simply saying taken to infinity the distribution of outcomes will tend to 50/50. > I still would not really understand what forces my coin to show tails eventually Nothing forces it to show heads either. > but if there were a measure showing that those two sets have measure 1 respectively 0 it would already be easier to swallow Well, we've already proven it's impossible to have all heads infinitely. The same would be true of any non-uniform sequence. Any sequence that favours heads by any margin at all, taken to infinity, would necessarily contain an infinite sub-sequence that contains all heads, which we've proven is impossible by the limit to infinity.
  - danbruc 10y ago
    Well, we've already proven it's impossible to have all heads infinitely. The same would be true of any non-uniform sequence. Any sequence that favours heads by any margin at all, taken to infinity, would necessarily contain an infinite sub-sequence that contains all heads, which we've proven is impossible by the limit to infinity. Have we? We only said that the probability of having an infinite sequence of all heads is zero but not that it is impossible to obtain that result. After all the probability of any infinite sequence is zero and therefore if we would equate probability zero with impossible to obtain, then we could obtain no infinite sequence at all. The second part is obviously wrong, H H T H H T... ad infinitum obviously favors heads over tails but does not contain an infinite subsequence of all heads.
- jerf 10y ago
  "Okay, assuming that is true, is there an intuitive way to understand that?" I mean this quite seriously and am not being sarcastic or dismissive: Probably not. I think the general concept of "intuitive" is that there is some experiential analog to the concept in the real world, with which we've had a lot of experience and can thus "intuit" the behavior. The real world does not include infinity. You can develop mathematical intuitions, but I don't think that's what you were saying. In the mathematical intuition sense, it's worth pointing out the probability of any given infinite series of coin flips is zero. The all-heads or all-tails series are not special that way. While this may not be the best way to intuit it, if a metaphysical you who would live forever sat down and started flipping coins, you will never at any point be done with flipping an infinite series of coins. You could sit there until you flip any finite series of coins, but you will never flip an infinite one, even with our unrealistic stipulations of life span. From that perspective, a probability of zero of flipping an infinite number of heads should seem reasonable; there is probability zero that you will ever have flipped an infinite number of coins.

danbruc 10y ago

Okay, assuming that is true, is there an intuitive way to understand that? And the fact that lim[n-> ∞] 0.5^n = 0 unfortunately won't do for me, that is true for any specific infinite sequence, even those containing 50/50 heads and tails. I think most of the sequences - hand-waving, most of an infinite set - are 50/50 heads and tails just because there are more possibilities - hand-waving again - to arrange 50/50 heads and tails versus 100/0 or 40/60 heads and tails. So can I sum over all the 50/50 sequences and get 1 and sum over the rest and get 0? I still would not really understand what forces my coin to show tails eventually, but if there were a measure showing that those two sets have measure 1 respectively 0 it would already be easier to swallow.

kareemsabri 10y ago
> Okay, assuming that is true, is there an intuitive way to understand that? And the fact that lim[n-> ∞] 0.5^n = 0 unfortunately won't do for me, that is true for any specific infinite sequence, even those containing 50/50 heads and tails. That's true, but only if you pick ONE specific sequence. The likelihood of any precise sequence of heads/tails is equally unlikely as all heads. However, we are not fixing the outcomes ordering, but simply saying taken to infinity the distribution of outcomes will tend to 50/50. > I still would not really understand what forces my coin to show tails eventually Nothing forces it to show heads either. > but if there were a measure showing that those two sets have measure 1 respectively 0 it would already be easier to swallow Well, we've already proven it's impossible to have all heads infinitely. The same would be true of any non-uniform sequence. Any sequence that favours heads by any margin at all, taken to infinity, would necessarily contain an infinite sub-sequence that contains all heads, which we've proven is impossible by the limit to infinity.
- danbruc 10y ago
  Well, we've already proven it's impossible to have all heads infinitely. The same would be true of any non-uniform sequence. Any sequence that favours heads by any margin at all, taken to infinity, would necessarily contain an infinite sub-sequence that contains all heads, which we've proven is impossible by the limit to infinity. Have we? We only said that the probability of having an infinite sequence of all heads is zero but not that it is impossible to obtain that result. After all the probability of any infinite sequence is zero and therefore if we would equate probability zero with impossible to obtain, then we could obtain no infinite sequence at all. The second part is obviously wrong, H H T H H T... ad infinitum obviously favors heads over tails but does not contain an infinite subsequence of all heads.
jerf 10y ago
"Okay, assuming that is true, is there an intuitive way to understand that?" I mean this quite seriously and am not being sarcastic or dismissive: Probably not. I think the general concept of "intuitive" is that there is some experiential analog to the concept in the real world, with which we've had a lot of experience and can thus "intuit" the behavior. The real world does not include infinity. You can develop mathematical intuitions, but I don't think that's what you were saying. In the mathematical intuition sense, it's worth pointing out the probability of any given infinite series of coin flips is zero. The all-heads or all-tails series are not special that way. While this may not be the best way to intuit it, if a metaphysical you who would live forever sat down and started flipping coins, you will never at any point be done with flipping an infinite series of coins. You could sit there until you flip any finite series of coins, but you will never flip an infinite one, even with our unrealistic stipulations of life span. From that perspective, a probability of zero of flipping an infinite number of heads should seem reasonable; there is probability zero that you will ever have flipped an infinite number of coins.