Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
&
Computer
Science
6.041/6.431:
Probabilistic
Systems
Analysis
(Fall
2010)
Problem
Set
3
Solutions
Due
September
29,
2010
1.
The hats of
n
persons are thrown into a box.
The persons then pick up their hats at random (i.e.,
so that
every assignment
of
the hats to the
persons is equally likely).
What is the
probability
that
(a)
every
person gets his or
her
hat
back?
Answer:
n
1
!
.
Solution:
consider
the sample space
of all
possible
hat assignments.
It has
n
!
elements (
n
hat
selections for
the first
person,
after
that
n
−
1 for
the
second,
etc.),
with every single
element
event
equally likely (hence having probability 1
/n
!).
The
question is to calculate
the probability of
a
singleelement event,
so the
answer
is 1
/n
!
(b)
the first
m
persons who
picked hats get their
own hats back?
(
n
−
m
)!
Answer:
.
n
!
Solution:
consider the same sample space and probability as in the solution of (a).
The
probability of
an event with (
n
−
m
)! elements (this is how many ways there
are
to disribute
the remaining
n
−
m
hats after
the
first
m
are
assigned to their
owners) is (
n
−
m
)!
/n
!
(c)
everyone among
the first
m
persons to pick up the
hats gets back a hat belonging to one
of
the last
m
persons to
pick up the
hats?
Answer:
m
!(
n
−
m
)!
=
n
1
=
n
1
. .
n
!
(
m
)
(
n
−
m
)
Solution:
there are
m
! ways to
distribute
m
hats among the
first
m
persons,
and (
n
−
m
)!
ways
to
distribute the remaining
n
−
m
hats.
The
probability of an event with
m
!(
n
−
m
)!
elements is
m
!(
n
−
m
)!
/n
!.
Now assume, in addition, that
every hat thrown into the
box has probability
p
of getting
dirty
(independently of
what
happens to
the
other
hats or
who has dropped or
picked it up).
What is
the probability that
(d)
the first
m
persons will pick up
clean hats?
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 Fall '10
 Prof.DimitriBertsekas
 Electrical Engineering, Probabilistic Systems Analysis, Massachusetts Institute, Department of Electrical Engineering & Computer Science

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