Chapter 4:
Probability
45
Chapter 4: Probability
4.1.
a.
The complement of event A is that it will take 4 days or less before the
machinery becomes operational.
b.
The intersection of A and B will be the event that it takes 5 days before the
machinery becomes operational.
c.
The union of A and B is the event of 1 day, 2 days, 3 days, 4 days, 5 days, 6
days or 7 days
d.
A and B are not mutually exclusive because P(A
∩
B)
≠
0
e.
Yes A and B are collectively exhaustive because they include all of the
possible sample points
f.
(A
∩
B) is the event that it takes 5 days.
(
A
∩
B) is the event that it takes 4
days, 3 days, 2 days, 1 day.
The union between these two events is that it
takes less than 6 days (1 through 5) before the machinery is operational.
This
is the definition of event B, therefore (A
∩
B)
∪
(
A
∩
B) = event B
g.
(
A
∩
B) is the event that it takes 4 days, 3 days, 2 days, 1 day.
Since A is the
event 5 days, 6 days, 7 days, Then A
∪
(
A
∩
B) will be the event of 1 through
7 days.
This is the event of A
∪
B.
Therefore, A
∪
(
A
∩
B) must equal A
∪
B.
4.2.
a.
(A
∩
B) is the event
that the Dow-Jones average rises on both days which is
O
1
.
(
A
∩
B) is the event the Dow-Jones average does not rise on the first day but
it rises on the second day which is O
3
.
The union between these two will be the
events O
1
O
3
which by definition is event B: the Dow-Jones average rises on the
second day.
b.
Since (
A
∩
B) is the event the Dow-Jones average does not rise on the first day
but rises on the second day which is O
3
and because A is the event that the Dow-
Jones average rises on the first day, then the union will be the event that either the
Dow-Jones average does not rise on the first day but rises on the second day or
the Dow-Jones average rises on the first day or both.
This is the definition of
A
∪
B.
4-3.
a.
Sample points in the sample space include the following 20 simple events:
M1,M2
M2,M1
M3,M1
T1,M1
T2,M1
M1,M3
M2,M3
M3,M2
T1,M2
T2,M2
M1,T1
M2,T1
M3,T1
T1,M3
T2,M3
M1,T2
M2,T2
M3,T2
T1,T2
T2,T1
b.
Event A is that at least one of the two cars selected is a Toyota
c.
Event B is that the two cars selected are the same model
d.
The complement of A is the event that the customers do not select at least one
Toyota
e.
(A
∩
B) is the event that at least one Toyota is selected and two cars of the
same model are selected.
Since (
A
∩
B) is the event that the customers do not
select at least one Toyota and two cars of the same model are selected, then