EECS 310, Fall 2011
Instructor: Nicole Immorlica
Problem Set #8
Due: November 22, 2011
1. (30 points) Let
S
=
{
1
,...,n
}
and let
π
be a random permutation on
S
(i.e.,
π
:
S
→
S
is a random bijection). Let
π
(
k
) represent the
k
th
element in the permutation
π
.
For a subset
A
of
S
let
f
(
A
) be the minimum of
π
(
a
) over all elements of
a
∈
A
. For
example, if
n
= 10,
A
=
{
2
,
5
,
7
}
, and
π
(2) = 3
,π
(5) = 10
,π
(7) = 2, then
f
(
A
) = 2. Let
A
and
B
be two arbitrary subsets of
S
. Express the following in terms of
A
,
B
, and
S
.
Be sure to explain your calculations.
(a) (6 points) What is the probability that
π
(1) = 1?
(b) (12 points) For an arbitrary element
a
∈
A
, what’s the probability that
f
(
A
) =
π
(
a
)?
(c) (12 points) What is the probability
f
(
A
) =
f
(
B
)?
2. (25 points) Consider the following hat puzzle: Three players enter a room wearing a hat.
Each hat is either red or blue, and the color of a hat is chosen uniformly at random. A
player can see every hat but his own. Each player must write on a card either
red
,
blue
,
or
pass
. After everyone writes down something, the cards are simultaneously revealed
(so no helpful information is revealed to the other players by what is written down on
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '11
 N/A
 Economics, Probability, Probability theory, Randomness, HT HT HHHT

Click to edit the document details