Mathematics - A First Course In Probability, 7Th Edition (Solutions Manual) - (Sheldon Ross) 2007.pdf

A Solution Manual for:
A First Course In Probability: Seventh Edition
by Sheldon M. Ross.
John L. Weatherwax∗
September 4, 2007
Introduction
Acknowledgements
Special thanks to Vincent Frost and Andrew Jones for helping find and correct various typos
in these solutions.
Miscellaneous Problems
The Crazy Passenger Problem
The following is known as the “crazy passenger problem” and is stated as follows. A line of
100 airline passengers is waiting to board the plane. They each hold a ticket to one of the 100
seats on that flight. (For convenience, let’s say that the k-th passenger in line has a ticket
for the seat number k.) Unfortunately, the first person in line is crazy, and will ignore the
seat number on their ticket, picking a random seat to occupy. All 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)?
If one tries to solve this problem with conditional probability it es very difficult. We
begin by considering the following cases if the first passenger sits in seat number 1, then all
∗******@
1
the remaining passengers will be in their correct seats and certainly the #100’th will also.
If he sits in the last seat #100, then certainly the last passenger cannot sit there (in fact he
will end up in seat #1). If he sits in any of the 98 seats between seats #1 and #100, say seat
k, then all the passengers with seat numbers 2, 3, . . . , k 1 will have empty seats and be able
to sit in their respective seats. When the passenger with− seat number k enters he will have
as possible seating choices seat #1, one of the seats k + 1, k + 2, . . . , 99, or seat #100. Thus
the options available to this passenger are the same options available to the first passenger.
That is if he sits in seat #1 the re