How would the world be different if only the WLLN were true?

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In summary, the conversation discusses the possibility of differentiating between the effects of the Weak Law of Large Numbers (WLLN) and the Strong Law of Large Numbers (SLLN) through a thought experiment involving flipping a coin. The speaker is unsure if it is possible to observe real world phenomena that would not be present if only the WLLN were true, and questions if it is possible to imagine a new set of assumptions for probability theory that would make one theorem true and the other not true.
  • #1
Poopsilon
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I imagine myself flipping a coin repeatedly and recording the outcomes. With only the WLLN being true, I expect to periodically encounter long strings of mostly heads or mostly tails, causing the running average to fall outside some epsilon's distance from the mean. These strings would occur with decreasing frequency, but with the possibility of another occurring always having positive probability.

If the SLLN were true I ought to instead expect that after reaching some number of coin flips, my running average would never again deviate outside of some epsilon's distance from the mean, however I'm not sure how many coin flips it will take for this to occur, and every time I think I might have passed this required number of coin flips associated to my particular choice of epsilon, if I do see another atypical string of mostly heads or mostly tails which takes my average outside of this epsilon, I just conclude that I actually hadn't yet reached the required number of coin flips after all.

Thus since I can only carry out a finite number of coin flips, I am unable to differentiate between the effects of the Weak Law and the hypothesized Strong Law.

I don't have the probabilistic/statistical expertise to carry out this thought experiment any further, nor to create a more nuanced one which might be capable of differentiating experimentally between the effects of the WLLN and the SLLN.

So I'm left wondering, is it possible to observe real world phenomena which would not be present if only the WLLN were true?
 
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It makes sense to ask how the world would be different of one of two assumptions were true and the other wasn't. But you are asking how the world would be different if one of two theorems were true and the other wasn't. The only way that could happen is if you can imagine a new set of assumptions for probability theory (and/or logic itself) that make one theorem true and the other not true. What would those assumptions be?
 

1. What is the WLLN?

The WLLN, or Weak Law of Large Numbers, is a mathematical theorem that describes the behavior of a large number of independent and identically distributed random variables. It states that as the number of observations increases, the average of those observations will converge to the expected value.

2. How would the world be different if only the WLLN were true?

If only the WLLN were true, it would have significant implications for the way we understand and make predictions about the world around us. The WLLN is a fundamental concept in statistics and probability, and its validity allows us to make accurate predictions about events based on large amounts of data.

3. Would all statistical models be based on the WLLN if it were the only true law?

No, the WLLN is just one of many statistical laws and principles that are used to understand and analyze data. While it is a powerful tool, it does not encompass all aspects of statistical modeling and would not be the only law used in this scenario.

4. How would the WLLN impact the way we collect and analyze data?

If the WLLN were the only true law, it would likely lead to a greater emphasis on collecting and analyzing large amounts of data. This is because the WLLN relies on having a large number of observations in order to accurately predict outcomes. It may also lead to more precise and reliable data analysis techniques being developed.

5. Are there any potential downsides to relying solely on the WLLN for making predictions?

While the WLLN is a powerful tool, it is not infallible. There may be situations where it does not accurately predict outcomes, and relying solely on it could lead to incorrect conclusions. Additionally, the WLLN assumes that the variables are independent and identically distributed, which may not always be the case in real-world scenarios.

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