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- Taking Seats on a Plane 8 answers
I came across a "brain teaser" which goes like this:
Bruna was first to arrive at a 100 seat theatre. She forgot her seat number and picks a random seat for herself.
After this, every single person who get to the theatre sits on his seat if its available else chooses any available seat at random. Neymar is last to enter the theatre and 99 seats were occupied.
What's the probability what Neymar gets to sit in his own seat?
The solution was $1/2$, their reasoning being that there are two possibilities, Neymar either gets his seat or not. This seems very flawed thinking to me because nothing says the two possibilities have equal probability.
I calculated it as $1/100 + 1/99$, but I think this is still wrong.
My reasoning was there are two scenarios which result in Neymar's seat being available, either:
- Bruna sits in the right seat in which case everyone also sits in their correct seat, this has a $1/100$ chance; or,
- Bruna sits in the wrong seat and there's a $98/99$ chance it is not Neymar's seat, then the next person has a $97/98$ chance of not sitting in Neymar's seat, and so on. Which gives $98!/99!$ or $1/99$.
My confusion lies in the fact that the subsequent people in the second scenario do not always take their seat at random, they only select a random seat if their seat is occupied.
How would I go about solving something like this?