I've come across this probability problem and could not solve it. In order to learn, I searched for the solution on the internet and found out that people just printed 1/2. i.e., the answer, no matter the value of n, is always 50%.
Why is that so? I tried making some DP for testing values under ~20 and only when n = 2 the answer is indeed 1/2.
Can someone give me some help on this tough (for me) one?
Appreciate the help!
The last vacant seat must be either seat 1 or seat n, because otherwise the person that owns it would have occupied it.
But because the first n-1 passengers don't have tickets for either one of those seats, there's no difference between those two seats in our problem. Therefore the probability that the last vacant seat is 1 is the same as the probability that the last vacant seat is n, and they equal 1/2.
Thank you!!!
Well, ffao was faster than me. Anyway, you could also think that if some person i < n sits in a wrong place, then there exists some j > i that will sit in a wrong place too.
Nice one too!! Thank you!!
Actually, don't think about it because it's wrong :p. In the way I described doesn't imply that the nth person will sit in a wrong place.