Not exactly, because the scenario isn’t “any two people on the plane” but “me and another person on the plane”.

We can see the full paradox at work here. The parent clearly states:

> how often people who know each other randomly run into each other in an airport.

> I mean, what are the odds?

This is explicitly stating "any two people" (and it's at an airport not a plane, so more people). But then follows up with changing the framing of question they're asking:

> I only had it happen once, but it was nuts.

The birthday paradox is only a paradox because we tend to think of birthdays in a very personal manner. So when we think of "any two people sharing a birthday" we immediately change this to "someone having my birthday", without realizing we've fundamentally changes the question we're asking.

The OP however isn’t surprised by the actual odds of any two people randomly running into each other, but by the fact that it happened to them or acquaintances. They actually don’t know what the odds would be for the whole airport or plane. With the birthday paradox, on the other hand, the thing that people are surprised about is not how often it happens to them, but the actual odds for a given group size of people.

Yes, but if it happened to someone else, that person might be on here expressing surprise and OP would be amusedly reading it.

I was in two classes where the birthday paradox was discussed, at different universities. Both times it was my birthday that was shared with someone else. What are the odds?

It is still not another specific person, but anyone you know, which is a pretty large set.

The birthday paradox is a paradox due to the quadratic odds. If you fix one person (yourself), the odds are only linear.