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I have read some good explanations on this site about the (weak and strong) laws of large numbers but I still have trouble applying the strong law of large numbers to the example of rolling a die.

I know that for a large number of rolls the average will converge to 3.5 almost surely. But what about the series that do not converge, like for example (6,6,6,6,...) or (5,6,5,6,5,6,...)? Why are such series still in line with the LLN? Is it because they almost surely never happen?

Edit: This is how I look at it after reading your answers. Let's say I would be able to write down all possible series of outcomes and put each of them in pot A if it converges to 3.5 or put it in pot B if it does not converge to 3.5. I would then notice that the probability of all the sets in pot A is 1 and that the probability of all the sets in pot B is 0. Is this correct?

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    $\begingroup$ Precisely. Convergence holds only outside some null set so it may not hold for a particular sequence of outcomes. $\endgroup$ – Kavi Rama Murthy Mar 4 '20 at 9:24
  • $\begingroup$ But what is the difference between the series in my question and a series S_n that converges. Don't all series have the same probability? $\endgroup$ – Florian Mar 4 '20 at 10:46
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The strong law of large numbers is about infinite sequences, so some may argue it is not related to the real world.

The weak law of large numbers, on the other hand, is about finite sequences, some1 say it may apply to the real world. For a given number $N$, the WLLN may give you an estimate of the probability that the outcomes of $N$ rolls of the dice have an average larger than $3.5 + \epsilon$. It quantifies our common sense notion that outcomes $(6,5,6,5,6,5,6,5,6,5,6,5)$ with a fair die would be "extremely unlikely".

1 (but not Littlewood)

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    $\begingroup$ Why not Littlewood? $\endgroup$ – Fede Poncio Mar 4 '20 at 14:26
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    $\begingroup$ Littlewood had a little essay "The Dilemma of Probability Theory" in his little collection, A Mathematician's Miscellany. He argued that any attempt to connect probability theory with the real world was flawed. $\endgroup$ – GEdgar Mar 4 '20 at 16:05
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Every single specific outcome $(a_1,a_2,\dots)$ has probability zero.

Still, if you consider a set of uncountably.many different outcomes, it might have probability zero or not, depending on the set.

The SLLN says that there is some set $N \subset \{1,\dots,6\}^\Bbb{N}$ which has probability zero and such that for all outcomes $(a_1,\dots)$ that do not belong to $N$, the convergence of $S_n$ to 3.5 takes place.

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  • $\begingroup$ Could you please give an example of a set that converges to 3.5 but does not have probability zero? $\endgroup$ – Florian Mar 4 '20 at 15:09
  • $\begingroup$ @Florian: It's not so easy to write such a set down explicitly. That's precisely the point of the SLLN, showing that the set of all "good" sequences has probability 1. If you are satisfied with a "non-explicit" set, you could for example take $\{1,\dots,6\} \setminus N$ where N is the null set from my answer. $\endgroup$ – PhoemueX Mar 4 '20 at 15:18
  • $\begingroup$ Would you suggest looking into the proof to get a better understanding of the set of all "good" sequences? $\endgroup$ – Florian Mar 5 '20 at 11:36

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