**Problem 10**

You are the last
person in a line of 100 people who are boarding an aircraft that has 100 seats.
Each person in line has a boarding pass assigning her/him to one of the seats
so that every seat is assigned to someone. The first person in line accidentally
drops his boarding pass into the ventilating system where it is immediately
shredded. Having forgotten his seat assignment, he then chooses a seat at
random and sits down. The remaining 99 people board the plane one at a time.
Each person takes his/her assigned seat if it is available and if it is not,
sits in one of the remaining open seats at random. What is the probability you
will end up sitting in your assigned seat?

[Problem
submitted by Roger Wolf, LACC Chairman of Mathematics.]

**Solution for Problem 10:**

**Solution 1**:

The nature of the
problem does not change if we renumber the seats so that the assigned seat
number corresponds to the person’s position in line. So the person whose ticket
was shredded was assigned seat number 1 and you were assigned seat number 100.

Consider changing the
seating process as follows: The person with the shredded ticket, call him Joe,
chooses a seat at random. The remaining 99 people board the plane one at a
time. Each person takes his/her assigned seat, bumping Joe out of the seat if
she/he finds him occupying it. Joe then immediately takes one of the remaining
open seats at random. Then the next person in line takes her/his seat, bumping
Joe if necessary, thus continuing the process until everyone is seated. It is
clear that the probability you do not have to bump Joe from your assigned seat
is the same as the probability you get your assigned seat in the original
problem.

In the newly
posed problem, let _{}be the event that, when you board the plane, Joe has been
bumped exactly *k* times. It is clear that _{}, and that _{}_{},_{},…..,_{} are mutually exclusive events such that =1. Let _{}be the event that the Joe never sits in your seat. The
solution to the problem is _{}. Clearly _{} for _{}. (When you board the plane, everyone but Joe is in his or
her assigned seat. Joe must be in seat #1 or your seat, #100. That means he sat
in his present location immediately after being displaced for the _{}time. If he had taken any other seat, it would have resulted
in a _{}displacement. The probability he sat in seat #1 given that he
chose randomly between seat #1 and seat #100, is _{}.)

Consequently:

_{}=_{}=_{}=_{}=_{}

**Solution 2**:

Let _{} be the nth guy in
line, _{}be the seat assigned to _{}. So _{} is assigned to _{} who lost his boarding
pass, and _{} is assigned to _{}, that is you. Let _{} be the event that
exactly *i* persons make a random choice of
seats. Given _{}, the last person who makes a random choice of seats must
choose his seat from {_{},_{}}. Therefore given_{}, the probability that you sit in your assigned seat is P(_{}→_{}∣_{})=1/2.

P(_{}→_{})=_{}(_{}→_{}∣_{}) = _{} = _{}_{}= _{}.