This problem I found on the following website. I am just restating it below
A line of 100 airline passengers is waiting to board a plane. They each hold a ticket to one of the 100 seats on that flight. (For convenience, let’s say that the nth passenger in line has a ticket for the seat number n.) Unfortunately, the first person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. All of the other passengers are quite normal, and will go to their proper seat unless it is already occupied. If it is occupied, they will then find a free seat to sit in, at random. What is the probability that the last (100th) person to board the plane will sit in their proper seat (#100)?
Now according to the solution provided by the website the answer is 0.5. Their reasoning is that the crazy situation in which people picking random chairs to sit ends when one of the non-crazy passenger who lost his seat picks the crazy first person’s seat. Hence the probability of the 100th person sitting on the right chair is the sum of probability of passenger 2 to 98 picking the first person’s chair
So that comes out according to them as
I feel the above reasoning is flawed. A passenger who lost his seat could pick the crazy first person’s seat but someone before him would have already picked the 100th person’s seat. I don’t think that scenario is taken into account in their reasoning.
So in my view the probability can be solved in reverse way as
In the above equation the notation
person sitting at
person not sitting at
The probability that the
person will not sit in his chair is the sum of the probability of any other person before him sitting in that chair. Therefore, this probability is given by
person did not get to sit in chair 98 but then ended up sitting on chair 100.
Using the reltationship
and since the crazy dude is always going to sit in someone else seat,
. The reason for
is because he has equal chance of picking one out of the 99 chairs ahead of him.
person when his seat is occupied and he has to pick a random seat from the available pool of
seats the chance of him picking any seat
are all equal and is given by
Expanding the problem to N seats
Now lets expand this problem to a scenario with N seats and the first person as previously is crazy and won’t sit in his chair and will select randomly a chair from 2 to N to sit. What is the probability that the last person gets to sit in his chair.
Using our previous reasoning the probability that
person won’t get to sit in his chair is given by
Since the first guy is not going to sit in his chair P(S_1\ne 1)=1$ and then he is going to choose any of the N-1 seats in random with probability given by
. For the
person, the probability of him sitting on
seat when his seat is occupied is given by
Similarly for the
person the probability that he won’t sit in his chair is
For the last guy the chance of him not sitting in his chair is when
or alternatively the probability that he will sit in the chair is
In the limiting case
I always like to verify my answer with a montecarlo simulation. The result for
seats/passengers with the first guy being crazy and the probability of the last guy getting his seat is shown below. The montecarlo simulation used 100,000 runs to determine the probability.