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
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 Fall '11
 N/A
 Economics, Probability, Probability theory, Randomness, HT HT HHHT

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