*Whenever the frst word oF a problem is pre
ceeded by an asterisk, the problem is For enrich
ment purposes only.
Chapter 1 Lecture Examples: FALL 2011
1. A CM has a sample space that consists oF
Four elements, denoted: a, b, c and d. As
suming the ELC, fnd the probabilities oF
each oF the Following events.
(a)
A
=
{
a
}
(b)
B
=
{
a, b
}
(c)
C
=
{
b, c, d
}
2. ReFer to the previous problem. Now, in
stead oF the ELC, assume that the probabil
ities oF a, b, c and d Follow the ratio 9:3:3:1.
(Note: IF interested, see
Mendelian inheri
tance
in Wikipedia For a discussion oF the
9:3:3:1 ratio, as well as the 1:2:1 and the
3:1 ratios.)
(a) Determine the probabilities oF the in
dividual outcomes a, b, c and d.
(b) Calculate the probabilities oF the
events
A
,
B
and
C
given in the pre
vious problem.
3. You are given the Following inFormation:
the events
A
and
B
are disjoint;
P
(
A
)=
0
.
40
;and
P
(
B
)=0
.
25
.Ca
lcu
la
tetheFo
l

lowing probabilities.
(a)
P
(
A
or
B
)
.
(b)
P
(
A
c
)
.
(c)
P
(
B
c
)
.
4. You are given the Following inFormation:
P
(
A
.
25
;
P
(
B
.
45
;
P
(
AB
0
.
20
te
P
(
A
or
B
)
.
5. What is wrong with each oF the Following?
(a)
P
(
A
.
20
;
P
(
B
.
55
;a
n
d
P
(
AB
.
25
.
(b)
P
(
A
.
60
;
P
(
B
.
55
A
and
B
are disjoint.
6. Consider a sample space with three mem
bers: 1, 2 and 3. Assume the ELC and i.i.d.
trials. The Following table helps to visual
ize the results oF the frst two trials:
X
2
X
1
1
2
3
1
(1,1)
(1,2)
(1,3)
2
(2,1)
(2,2)
(2,3)
3
(3,1)
(3,2)
(3,3)
The nine entries in this table are equally
likely.
Defne
X
=
X
1
+
X
2
,thetotaloFthenum
bers obtained in the frst two trials. ±ind the
sampling distribution oF
X
.
7. Consider a sample space with fve mem
bers: 0, 1, 2, 3 and 4. Assume the ELC
and i.i.d. trials. The Following table helps
to visualize the results oF the frst two trials:
X
2
X
1
0
1
2
3
4
0
(0,0)
(0,1)
(0,2)
(0,3)
(0,4)
1
(0,1)
(1,1)
(1,2)
(1,3)
(1,4)
2
(0,2)
(2,1)
(2,2)
(2,3)
(2,4)
3
(0,3)
(3,1)
(3,2)
(3,3)
(3,4)
4
(0,4)
(4,1)
(4,2)
(4,3)
(4,4)
The 25 entries in this table are equally
likely.
Defne
X
=
X
1
X
2
,theproductoFthenum
bers obtained in the frst two trials. ±ind the
sampling distribution oF
X
.
1