That actually happened to my Dad (who also happens to be a particle physicist). He and another professor were working some problem in their office lounge when some snot-nosed undergrad wandered in, looked at the board for about 2 minutes, and - completely ignoring the two professors - walked over, and corrected it.

He ended up being one of Dad's grad students.

Dad's also a Civil War historian (with an emphasis on the Mississsippi River campaigns). Knowing him for my entire life, I've never seen him arrogant - though I have seen him slice people off at the ankles. I've never knwn him to offer an opinion on a subject he has not studied - at least to some extent.

He's certainly willing to admit being wrong - but he will also demand you prove your assertions. I suppose that might be seen as arrogant...