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CEE 373
Homework #2
Due date: September 15 at 5pm
Total: 100 points
1. (20 points) A General Contractor is managing three different renovation projects
in the Metro Detroit Area. Let
A
i
denote the event that renovation project at
site
i
is completed by the contract date.
For each of the following events, (1) describe the event as set operations in terms
of
A
1
,
A
2
, and
A
3
using union, intersection, and complementation notation and
(2) draw a Venn diagram shading the region that describes the event.
(a) Two projects are completed by the contract date.
(b) Three projects are not completed by the contract date.
(c) Only the project at site 1 is completed by the contract date.
(d) Exactly one project is completed by the contract date.
(e) Either the project at site 1 or both of the other two projects are completed
by the contract date.
Solution:
(a) (
A
1
∩
A
2
∩
A
3
)
∪
(
A
1
∩
A
2
∩
A
3
)
∪
(
A
1
∩
A
2
∩
A
3
)
(b)
A
1
∪
A
2
∪
A
3
Page 1 of 10
CEE 373
Homework #2
(c)
A
1
∩
A
2
∩
A
3
(d) (
A
1
∩
A
2
∩
A
3
)
∪
(
A
1
∩
A
2
∩
A
3
)
∪
(
A
1
∩
A
2
∩
A
3
)
(e)
A
1
∪
(
A
2
∩
A
3
)
2. (20 points) Assume that you are a design engineer in Puerto Rico, where the
probability of a high wind occurring in any single minute is 10
−
7
, and the proba-
bility of a strong earthquake occurring in any single minute is 10
−
5
. Assume that
the occurrence of high winds and strong earthquakes is independent.
Page 2 of 10
CEE 373
Homework #2
(a) What is the probability of both high wind and a strong earthquake occurring
in the same minute in Puerto Rico?
(b) What is the probability of one or the other events occurring in the same
minute?
(c) Building codes do not require you as an engineer to design buildings for the
combined effect of wind and earthquake. Is this reasonable?
(d) For rare events, engineers often assume that
P
(
E
1
∪
E
2
)
∼
=
P
(
E
1
) +
P
(
E
2
).
Is this a reasonable assumption?
(e) If the occurrence of events in successive minutes is mutually independent,
what is the probability that there will be no high winds in Puerto Rico this
year? What is the probability that there will be no high winds in Puerto
Rico in 10 years?
Note: while high winds and an earthquake didn’t occur in the same minute, Puerto
Rico faced two hurricanes (Irma and Maria) in 2017, earthquakes in 2019, and
the COVID-19 pandemic in 2019), from which they are still recovering.
Solution:
Denote occurrence of high wind as
E
1
and the occurrence of strong
earthquake as
E
2
(a)
P
(
E
1
∩
E
2
) =
P
(
E
1
)
P
(
E
2
) = 10
−
5
·
10
−
7
= 10
−
12
(As
E
1
and
E
2
and inde-
pendent)
(b)
P
(
E
1
∪
E
2
) =
P
(
E
1
) +
P
(
E
2
)
−
P
(
E
1
E
2
) = 10
−
5
+ 10
−
7
−
10
−
12
≃
1
.
01
∗
10
−
5
(c) If both events are very rare the probability of their joint occurrence is negligi-
ble, so neglecting the joint effect of earthquake and winds is reasonable.
(d)
P
(
E
1
∪
E
2
) =
P
(
E
1
) +
P
(
E
2
)
−
P
(
E
1
E
2
).
For rare events the probability
P
(
E
1
E
2
) is negligible so it can be neglected leading to the formula
P
(
E
1
∪
E
2
)
≃
P
(
E
1
)+
P
(
E
2
). When dealing with probabilities of hazardous situations
this leads to over estimation of probability which is conservative.
If the probabilities are not small, however, then the approximation is not
accurate and in some cases can even lead to probabilities greater than 1.
(e)
P
(no high wind in a year) =
P
(
¯
E
1
)
365
∗
24
∗
60
= (1
−
10
−
7
)
525600
= 0
.
949
P
(no high wind in 10 years) =
P
(
¯
E
1
)
365
∗
24
∗
60
∗
10
= (1
−
10
−
7
)
5256000
= 0
.
591
Page 3 of 10
CEE 373
Homework #2
3. (15 points) Consider the transportation network below. Links
A
-
E
connect nodes
1-4. Each link may be “open” or “closed.” If a link is “open,” it is possible to
travel across that link, and if it is “closed,” it is impossible.
Let’s define event
E
A
as the event in which link
A
is open, and we define events
E
B
, E
C
, ...E
E
analogously.
(Let
¯
E
A
...
¯
E
E
be the complements of these events.)
We can now define other events in terms of these “link” events. For example, let
E
14
be the event in which a path is open between nodes 1 and 4. Then we may
say that:
E
14
= (
E
A
∩
E
D
)
∪
(
E
E
∩
E
C
)
∪
(
E
E
∩
E
B
∩
E
D
)
∪
(
E
A
∩
E
B
∩
E
C
)
For each of the cases below, find a similar set-theoretic expression for the event
described, in terms of events
E
A
....
E
E
together with their complements.
(a)
E
13
, the event in which an open path exists between nodes 1 and 3
(b)
¯
E
13
, the event in which there is no open path between nodes 1 and 3
(c)
E
13
∩
¯
E
34
, the event in which an open path exists between nodes 1 and 3 but
not between 3 and 4
Note: In each case, your final answers need to have the following characteristics:
Distribute all intersections over unions. For example:
(
E
A
∪
E
B
)
∩
¯
E
C
=
(
E
A
∩
¯
E
C
)
∪
(
E
B
∩
¯
E
C
)
Use the fact that unions are associative to eliminate parentheses. For example:
(
E
A
∪
E
B
)
∪
¯
E
C
=
E
A
∪
(
E
B
∪
¯
E
C
)
=
E
A
∪
E
B
∪
¯
E
C
Make use of mutual exclusivity wherever possible, to simplify; i.e.,
E
B
∪
(
E
C
∩
¯
E
C
)
=
E
B
∪ ∅
=
E
B
Make use of complementarity wherever possible, to simplify; i.e.,
E
A
∩
(
E
C
∪
¯
E
C
)
=
E
A
∩
S
=
E
A
Eliminate all compound complements; i.e.,
E
A
∪
E
B
=
¯
E
A
∩
¯
E
B
E
A
∩
E
B
=
¯
E
A
∪
¯
E
B
Page 4 of 10
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