From one point of view, this may sound a little silly -- a proof is a proof, right? Once you've proved something, what else is there to say? But sometimes, rather than just saying "Okay, problem solved, so I just throw away my original intuition and never think of it again", we feel the need to "work on" that original intuition -- how do I come up with a new informal, imprecise, intuitive way of looking at things in order to try to convince my intuition that the correct answer is Q?
Here's an example that sort of qualifies for me.
Consider n = 3.
There are 26^3 possible words. How many of them contain XY as a subword? There are 26 words of the form XY*, and 26 of the form *XY. So the probability of containing XY as a subword is 2*26/26^3.
How many words contain XX as a subword? There are 26 words of the form XX*. There are 26 of the form *XX, but there are 2*26-1 such words altogether, because we counted the word XXX twice.
So (at least for words of length 3) for a randomly generated word, the probability of containing XY as a subword is slightly higher than the probability of containing XX as a subword. (As it turns out, that trend continues for larger n.)
Roughly paraphrasing the argument, XY is more likely because it can't overlap with itself.
At first, I had a really hard time wrapping my mind around that and making it intuitive. How on earth does the fact that the word doesn't overlap with itself make it "more likely"? It almost seemed like, if anything, the word that can overlap with itself should be "more likely".
I wrestled with this for a little while and managed to change my intuition a bit. I can maybe post some remarks later about that, but I thought there was a chance that people here might want to wrestle with this one. And I hope you think the question in general is an interesting one -- what are some tricks for changing your intuition so it conforms with the facts? Can you think of other interesting examples from your own experience?