I always find the strong law of large numbers hard to motivate to students, especially non-mathematicians. The weak law (giving convergence in probability) is so much easier to prove; why is it worth so much trouble to upgrade the conclusion to almost sure convergence?


I think it comes down to not having a good sense of why, practically speaking, a.s. convergence is better than convergence i.p. Sure, I can prove that one implies the other and not conversely, but the counterexamples feel contrived. I understand the advantages of a.s. convergence on a technical level, but not on the level of everyday life.


So my question: how would you explain to, say, an engineer, the significance of having a.s. convergence as opposed to i.p.? Is there a "real-life" example of bad behavior that we're ruling out?



Motivation for strong law of large numbers
[Here](http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/) is a nice post of T. Tao on SLLN. In the comments section he is asked a very similar question to which he answers the following: (I hope it's ok to reproduce it here, since it is buried down in the comments)



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> Returning specifically to the question of finitary interpretations of the SLLN, these basically have to do with the situation in which one is simultaneously considering multiple averages $\overline{X}\_n$ of a single series of empirical samples, as opposed to considering just a single such average (which is basically the situation covered by the WLLN). For instance, if one had some random intensity field of grayscale pixels, and wanted to compare the average intensities at 10 x 10 blocks, 100 x 100 blocks, and 1000 x 1000 blocks, then the SLLN suggests that these intensities would be likely to be simultaneously close to the average intensity. (The WLLN only suggests that each of these spatial averages are individually likely to be close to the average intensity, but does not preclude the possibility that when one considers multiple such spatial averages at once, that a few outlying spatial averages will deviate from the average intensity. In my example with only three different averages, there isn’t much difference here, as the union bound only loses a factor of three at most for the failure probability, but the SLLN begins to show its strength over the WLLN when one is considering a very large number of averages at once.)
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