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

[Poopsilon]

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?

 

[Stephen Tashi]

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?