TITLE: Motivation for strong law of large numbers
QUESTION [21 upvotes]: 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?

REPLY [7 votes]: Suppose you are in the context of collecting data and estimating the mean.
Imagine a situation where SLLN does not hold. It means that with positive probability accumulating new data is useless.