What’s the probability that a random n-digit palindrome is divisible by 11?

What I find interesting is that the question sets you up for finding something interesting, just like the recent "what are the chances that a random number contains the digit 3" (the answer was "the same as finding any other digit and this approaches 1 as the number gets larger e.g has more digits" - a not remarkable insight at all).

However, in a classroom setting, or anywhere one is doing maths for fun, this is a great idea - it requires good problem solving techniques to find the answer and it is the process not the outcome that is the game.