AMS 311 (Fall, 2011)
Joe Mitchell
PROBABILITY THEORY
Homework Set # 1 – Solution Notes
(1).
(13 points)
A system is comprised of 5 components, each of which is either working or failed. Consider
an experiment that consists of observing the status of each component, and let the outcome of the experiment
be given by the vector
(
x
1
,x
2
,x
3
,x
4
,x
5
)
, where
x
i
is equal to 1 if component
i
is working and is equal to 0
if component
i
is failed. (a). How many outcomes are in the sample space of this experiment? (b). Suppose
that the system will work if components 1 and 2 are both working, or if components 3 and 4 are working, or
if components 1, 3, and 5 are all working. Let
W
be the event that the system will work. Specify all of the
outcomes in
W
. (c). Let
A
be the event that components 4 and 5 are both failed. How many outcomes are
contained in the event
A
? (d). Write out all the outcomes in the event
A
∩
W
.
(a). Since each
x
i
is either 0 or 1 (2 choices), the total number of choices for the outcome vector,
(
x
1
,x
2
,x
3
,x
4
,x
5
), is 2
5
. Thus,

S

= 2
5
= 32.
(b).
W
has 15 elements, namely:
W
=
{
(1,1,0,0,0), (1,1,0,0,1), (1,1,0,1,0), (1,1,0,1,1), (1,1,1,0,0),
(1,1,1,0,1), (1,1,1,1,0), (1,1,1,1,1), (0,0,1,1,0), (0,0,1,1,1), (0,1,1,1,0), (0,1,1,1,1), (1,0,1,1,0), (1,0,1,1,1), (1,0,1,0,1)
}
.
(c). Since
A
consists of outcomes having
x
4
=
x
5
= 0, we see that there are 2 choices for each of
x
1
,x
2
,x
3
;
thus,
A
has 2
·
2
·
2 = 8 elements. (You should be able to list them.)
(d).
A
∩
W
=
{
(1
,
1
,
0
,
0
,
0)
,
(1
,
1
,
1
,
0
,
0)
}
, since these are the only two elements of
W
having
x
4
=
x
5
= 0.
(2).
(16 points)
Sixty percent of the students at a certain school wear neither a ring nor a necklace. Twenty
percent wear a ring and 30 percent wear a necklace. If one of the students is chosen randomly, what is the
probability that this student is wearing (a). a ring or a necklace? (b). a ring and a necklace? Begin by Frst
describing exactly what the sample space is, and give names to the events (e.g., “
R
” and “
N
”).
The experiment is the selection of one student at random from the school. The sample space is the set