[psyang]

Originally Posted by GGG

Spoiler!

I think you are right with the wrong logic

The question is does the jack asses seat or your seat get filled first. Anyone who is sitting in their own seat doesn’t matter.

So person sits in own seat, doesn’t matter nothing changes. Jackass sits down equal prob of him having his seat or your seat. If neither occurs we created a new Jack ass. Eventually new Jack ass gets on and again is 50-50 of sitting in the Jack ass or your seat. If neither of those conditions a new jackass is created.

We have a winner!

The formal proof might go something like:

Spoiler!

Let's call the troublemaker T (I like the evolution of T from illiterate->wrong seat selector->dumb passenger->Jackass!)

Proof by induction.

Let's say T is the 99th passenger or (for our numbering scheme) 1 ahead of your position. Then T can choose either his seat or your seat. Probability of you getting your seat when T is 1 ahead of you is:

P(1) = 1/2

We now assume that

P(k) = 1/2 for k<=n

What if T is n+1 ahead of you?

Then P(n+1) = T choosing his seat OR T choosing any other passenger's seat except for yours (making that person the new T).
=1/(n+2) + n/(n+2)*P(k) where k <= n
=1/(n+2) + n/(n+2)*(1/2) (P(k)=1/2 when k<=n by our induction assumption)
=2/(2n+4) + n/(2n+4)
=(n+2)/(2n+4)
=1/2

I like the formalism of the above proof, but honestly, I like GGG's descriptive proof better because it is more obvious what is going on.