Homework #1
CIVL 7012/8012
Due Wednesday, January 28, 2009
1. Three events are shown on the Venn diagram in the following figure. Reproduce
the figure and shade the region that corresponds to each of the following events:
a.)A′
b.)A ∩ B
A
B
c.)(A ∩ B) C
d.)(B  C) ′
e.)(A ∩ B)′ C
C
2. A part selected for testing is equally likely to have been produced on any one of
six cutting tools.
a. What is the sample space?
b. What is the probability that the part is from tool 1?
c. What is the probability that the part is from tool 3 or tool 5?
d. What is the probability that the part is not from tool 4?
3. The analysis of shafts for a compressor is summarized by conformance to
specifications.
a. If a shaft is selected at random, what is the probability that the shaft
conforms to surface finish requirements?
b. What is the probability that the selected shaft conforms to surface finish
requirements or to roundness requirements?
roundness conforms
yes
no
surface finish
yes
345
5
conforms
no
12
8
4. The probability that a lab specimen contains high levels of contamination is 0.10.
Five samples are checked, and the samples are independent.
a. What is the probability that none contains high levels of contamination?
b. What is the probability that exactly one contains high levels of
contamination?
c. What is the probability that at least one contains high levels of
contamination?
5. A steel plate contains 20 bolts. Assume that 5 bolts are not torqued to the proper
limit. Four bolts are selected at random, without replacement, to be checked for
torque.
a. What is the probability that all four of the selected bolts are torqued to the
proper limit?
b. What is the probability that at least one of the selected bolts is not torqued
to the proper limit?
6. A biotechnology manufacturing firm can produce diagnostic test kits at a cost of
$20. Each kit for which there is a demand in the week of production can be sold
for $100. However, the half-life of components in the kit requires the kit to be
scrapped if it is not sold in the week of production. The cost of scrapping the kit
is $5. The weekly demand is summarized as follows:
Weekly demand
Number of units
0
50
100
200
Probability of demand
0.05 0.4
0.3
0.25
How many kits should be produced each week to maximize the mean earnings of
the firm?
CIVL 8012
Work all eight problems above, and add the following:
7. Suppose that a lot of washers is large enough that it can be assumed that the
sampling is done with replacement. Assume that 60% of the washers exceed the
target thickness.
a. What is the minimum number of washers that need to be selected so that
the probability that all the washers are thinner than the target is less than
0.10?
b. What is the minimum number of washers that need to be selected so that
the probability that one or more washers are thicker than the target is at
least 0.09?