University
Problem Set 3 ECE 313
of Illinois Fall 1998
Assigned :
Wednesday, September 9, 1998
Due :
Wednesday, September 16, 1998
Reading :
Ross, Chapter 2, Chapter 3.2
1. Let
A; B; C
denote three events dened on the same sample space. Prove that
(a) (
P
(
A
)+
P
(
B
)+
P
(
C
))
=
3
P
(
A
[
B
[
C
)
P
(
A
)+
P
(
B
)+
P
(
C
)
:
(b) Bonferroni's inequality:
P
(
ABC
)
P
(
A
)+
P
(
B
)+
P
(
C
)
,
2
:
(c) Generalized Bonferroni's inequality: Given
n
events
A
1
;A
2
;::: ;A
n
, use induction to show
that
P
(
A
1
A
2
:::A
n
)
P
(
A
1
)+
P
(
A
2
)+
P
(
A
n
)
,
(
n
,
1)
:
2. Find
P
(
A
[
(
B
c
[
C
c
)
c
) in each of the following cases:
(a)
A; B; C
are mutually exclusiveevents and
P
(
A
)=3
=
7.
(b)
P
(
A
)=1
=
2
;P
(
BC
)=1
=
3
;P
(
AC
)=0
:
(c)
P
(
A
c
B
c
[
A
c
C
c
)=3
=
7
:
3. Ross Chapter 2, p. 60, Problem 43.
4. Ross Chapter 2, p. 61, Problem 56.
5. (a) Compute the probability that a bridge hand (13 cards) is void (no card) in at least one suit
(Spades, Clubs, Hearts and Diamonds).
(b) Compute the probability that a bridge hand has at least one card from each suit (Spades,
Clubs, Hearts and Diamonds).
6.
Matching Problem [15 pts]:
A router receives 8 packets that need to be routed to stations
A; B; C; D; E; F
,
G
, and
H
, each station receiving exactly one packet. Due to a lightning strikeat
the router site, it cannot parse the header information in a packet and hence, with equal probability,
sends a given packet to any one of the 8 stations.
(a) Find the probability that the router redirects all packets wrongly.
(b) Find the probability that the router gets at least 6 packets routed correctly.
(c) Given that Station A does
not
receive the correct packet, what is the probability that
none
of
the stations receive their designated packets?
1
