Flip two coins and walk a mile north for each heads, and a mile south for each tails.
How much is your average cab-fare home?
Phrased this way, we know the answer isn't "zero" because no cabby is going to pay you for walking south instead of north. Doing the math, a quarter of the time you end up 2-miles north, a quarter of the time you end up 2-miles south, and the other half the times you've already made it home; altogether that amounts to 1-mile worth of cab-fare on average. That's not so hard, right?
So what's with the stupid "How far away am I, on average, from where I started?... (Zero is wrong...)" wording? If they didn't want to know 'where' I end up on average, they should have asked 'how far'... umm, I mean... well, that's just annoying now isn't it.
And that's why it was worded that way! The lesson isn't supposed to be "And now class, we define a single pedantic definition of what 'How far away?' is supposed to mean!" but rather the lesson is that everyday language can sometimes be vague and/or imprecise.
Or at least, that's the usual place where I see this example: in distance-versus-displacement pedagogy.
Connecting this example to "misunderstanding probability" is somewhat unusual, but it's not entirely incorrect either. In one way it illustrates how, just because you can crunch numbers and output a statistic doesn't necessarily mean that you have actually produced the useful data that you think you wanted.
How so? Well, we've determined the average displacement and average distance for this example: on average you end up back where you started, but on average you spend 1-mile worth of cab-fare. If we didn't just work through the math and know exactly what each of those two statistics signified and how they do/don't relate, we might have looked at those numbers and thought... "Why am I paying a cabby for 1-miles on average when I'm 0-miles from home on average?" or even worse "Why on Earth am I paying somebody to drive me 1-mile away from home!?". Both conclusions are total nonsense and we know why (having done all that the math above) but these kinds of conclusions get made quite easily. Knowing that calculating average cab-fare required the average-distance statistic (rather than assuming that average-displacement was the 'same thing') was vital to understanding the whole situation.
Similarly, problems involving combinatorics, probabilities, and statistics will rely on a chain of logical steps being accounted for properly. What if we hadn't properly accounted for TH and HT separately? We would have thought our average cab ride was (2+0+2)/3=1.33-miles long! Probability problems can be notoriously tricky for new students precisely because it can be hard to properly account for every possibility when you barely know what adding versus multiplying probabilities means and you're just winging it.