A/the fair

Hi, can someone please confirm if my analysis of this is right?

Let's say I see a Christmas fair down the street from my school. I go to my school and ask another student if he has seen the fair as well.

"Joel, have you seen A fair down the street?" or "Joel, have you seen THE fair down the street?"

My analysis: "a fair" is correct, but it seems a bit unnatural here. It's almost as if I am doubting my own eyes (that I have seen it myself). "The fair" suggests that I think there is a high probability that he has seen it. It's very specific (THAT fair down the street). Basically I'd use "the" here, although "a" is not incorrect. Is the analysis correct?