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?