98 2. FUNDAMENTAL RELIABILITY MATHEMATICS
2.15 EXERCISES
2.1. What is an experiment? What is an event? Use one example to explain them.
2.2. What is the sample space? What is the sample point? Use one example to explain them.
2.3. For an experiment of throwing a dice, the two sets are A D f1; 2; 3g, B D f1; 3; 4g. De-
termine A [ B and A B.
2.4. Toss a coin for a total of 1000 times. e occurrence of “head” was 489. What is the
probability of occurrence of the “head”? What is the probability of occurrence of the
“tail”?
2.5. When two dice are rolled, determine the probability of realizing the sum of two numbers
less than six.
2.6. A box contains four white balls and five green balls. A ball is selected randomly from
the box, and its color is noted. e ball is then returned to the box, and another ball is
selected randomly, and its color is noted. Determine the probabilities of realizing the
following events:
(a) two red balls, and
(b) first ball red and the second ball green.
2.7. Describe and explain a random event and a random variable. List two examples for each
in mechanical designs.
2.8. For an experiment of rolling a dice, D
f
1; 2; 3; 4; 5; 6
g
, A D
f
1; 2; 3
g
, B D
f
3; 4; 5; 6
g
, C D f5; 6g. Determine the probabilities: P
.
A [B
/
, P
.
A B
/
, P
.
A [C
/
,
P
.
A C
/
.
2.9. An experiment consists of tossing a coin four times.
(a) Find the PMF and distribution function for the number of heads realized.
(b) Find the probability of realizing heads at least two times.
2.10. A small village has two grocery stores A and B. It is estimated that 40% of potential
customers do business only with the store A, 30% only with store B, and 20% with both
stores A and B. e remaining 10% of customers do business with none of the stores
A and B. If E
1
.E
2
/ denote the event of a randomly selected customer doing business
with the store A (B), calculate the probabilities: P .E
1
[ E
2
/, P .E
1
E
2
/, P .E
1
/, and
P . E
1
[ E
2
/.
2.11. A batch of 195 specimens of composite material has the data shown in Table 2.15 when
they are tested for strength and density.
2.15. EXERCISES 99
Table 2.15: Strength and density
High Density Low Density
High Strength 142 22
Low Strength 16 15
(a) For a specimen randomly selected from the batch, calculate the probability that
both its strength and density are low.
(b) For a specimen randomly selected from the batch, calculate the probability that its
strength is low, but density is high.
(c) Let E
1
denotes that a specimen has high strength and E
2
denotes that a specimen
has a high density. Are the events E
1
and E
2
independent to each other?
2.12. e inspection of a batch 1000 castings produced in an engine production plant for the
presence of defect yielded the data in Table 2.16.
Table 2.16: Presence of defect
Type of defect
Location of Defect:
Inside
Location of Defect:
On the surface
Total
Blowholes
600 230 830
Crack 75 95 170
Total 675 325 1000
For a casting selected randomly from the batch, determine the following.
(a) e probability that the casting has a crack.
(b) e probability that the casting has a defect on the surface.
(c) e probability that the case has a blowhole which is on the surface.
(d) e probability that the casting has a crack which is inside.
2.13. An applied mathematics class consists of 35 male and 15 female students. Among the
male students, 15 are science majors, and 20 are engineering majors. Among the female
students, 5 are science majors and 10 are engineering majors. E
1
denotes a female stu-
dent and E
2
a male student. M
1
denotes a student in a science major and M
2
a student
in an engineering major.
(a) Calculate P .E
1
j
M
1
/.
100 2. FUNDAMENTAL RELIABILITY MATHEMATICS
(b) Calculate P .M
2
j
E
2
/.
(c) Calculate P .E
1
M
1
/.
(d) Calculate P .E
2
M
1
/.
2.14. In a thermal power plant, four components—the feed water pump, boiler, turbine, and
generator—are connected one after another as shown in the following figure. e prob-
abilities of each component are also listed in Figure 2.30. It is assumed that the failure
of any component is independent of each other. Calculate the probability that all four
components are functional.
Pump
R = 0.92
Boiler
R = 0.98
Turbine
R = 0.99
Generator
R = 0.95
Figure 2.30: Schematic of the block diagram for Problem 2.14.
2.15. In an engine manufacturing plant, 70% of the crankshafts are ground by a machin-
ist R and 30% by another machinist S . It is known from previous experience that the
crankshafts ground by the machinists R and S contain 2% and 3% defective units, re-
spectively. What is the probability of defective units?
2.16. A batch of 1000 piston O-ring manufactured in a factory contains 35 defective ones.
Two piston rings are randomly selected from the batch, one at a time, without replace-
ment. E
1
denotes a defective O-ring in the first selection and E
2
a defective O-ring in
the second selection.
(a) Calculate the probability P .E
1
/.
(b) Calculate the probability P .E
2
/.
(c) Calculate the probability P .E
1
E
2
/.
2.17. e percentage of mechanical, civil, biomedical engineering students in an Engineer-
ing Statics course is 37, 40, and 23, respectively. e probability of a mechanical, civil,
biomedical engineering student getting the grade A is 0.22, 0.15, and 0.32, respectively.
(a) Calculate the probability of a student getting grade A in this class.
(b) If a student is randomly selected and has a grade A, calculate the probability of the
student belonging to mechanical engineering.
2.18. Describe and explain mean, standard deviation, and coefficient of variance.
2.19. e diameter X of manufactured shafts (unit in inch) by two operators are listed in
Table 2.17.
2.15. EXERCISES 101
Table 2.17: Manufactured shafts
Operator A 2.99, 3.21, 3.33, 3.31, 3.38, 3.29, 3.25, 3.08
Operator B 3.01, 2.89, 3.45, 3.89, 2.76, 3.34, 3.01, 3.49
(a) Calculate the mean and standard deviation of the diameter by each operator.
(b) Which of the operators is better based on the data? Why?
2.20. Tensile test data of 100 specimens of a material are listed in Table 2.18.
Table 2.18: Tensile test data
Yield Strength Data
55.8, 50.3, 59.4, 56.4, 55.6, 57.4, 56.2, 50.7, 54.3, 54.3, 52.6, 62.7, 51.0, 52.8, 51.7, 49.6, 54.1, 53.7, 62.0, 53.9,
50.1, 62.4, 60.7, 53.9, 48.1, 53.0, 54.3, 56.3, 53.8, 57.0, 56.8, 49.2, 50.6, 51.6, 52.7, 53.5, 55.1, 41.1, 52.9, 60.7,
50.1, 59.3, 56.6, 54.9, 55.8, 47.8, 54.6, 62.4, 55.5, 55.2, 51.6, 54.9, 56.1, 57.0, 53.3, 53.9, 64.3, 44.6, 65.3, 56.6,
59.6, 47.3, 52.3, 53.7, 56.9, 47.3, 57.2, 49.4, 55.3, 58.0, 56.5, 60.0, 59.6, 52.0, 56.2, 50.7, 48.9, 59.3, 55.0, 54.7,
59.2, 57.7, 56.6, 60.8, 59.3, 56.1, 51.8, 52.0, 60.5, 47.6, 54.9, 46.0, 59.7, 59.0, 55.0, 54.7, 43.6, 57.7, 44.9, 44.3
(a) Use six equal non-overlapping bin intervals and manually count the number of test
data in each bin interval and draw the histogram.
(b) Input the data into one column in an Excel file and then use the histogram tool in
Excel to draw the histogram with ten equal bin intervals.
(c) Compile a MATLAB program to read the data through the Excel file and use the
histogram command to draw the histogram with 11 bins.
2.21. One hundred sampling data of a random variable are listed in Table 2.19.
Table 2.19: Random variable
e Sampling Data
41.2, 25.7, 34.9, 60.2,35.2, 19.7, 21.6, 30.4, 30.1, 21.5, 36.2, 62.4, 29.2, 29.9, 11.0, 22.5, 16.5, 27.3, 43.1, 70.9,
81.4, 31.2, 28.2, 49.8, 42.3, 48.6, 48.9, 15.8, 43.4, 31.6, 22.6, 23.4, 62.8, 22.1, 17.8, 36.9, 90.5, 33.5, 38.6, 20.7,
76.5, 35.3, 43.2, 20.6, 50.8, 40.2, 18.6, 33.8, 26.4, 35.0, 29.2, 30.1, 19.8, 28.2, 48.6, 79.2, 18.5, 71.0 ,36.0, 28.5,
23.4, 50.2, 23.4, 26.3, 51.5, 41.2, 51.9, 42.6, 27.1, 25.6, 49.3, 23.7, 59.9, 49.2, 38.2, 33.2, 39.8, 19.1, 31.3, 15.2,
86.3, 108.7, 44.9, 24.0, 122.6, 43.6, 38.4, 22.4, 19.4, 13.7, 26.8, 19.6, 45.8, 28.3, 80.2, 70.5, 35.9, 28.7, 58.9, 18.7,
102 2. FUNDAMENTAL RELIABILITY MATHEMATICS
(a) Use ten equal non-overlapping bin intervals and manually count the number of
test data in each bin interval and draw the histogram.
(b) Input the data into one column in an Excel file and then use the histogram tool in
Excel to draw the histogram with ten equal bin intervals.
(c) Create a MATLAB program to read the data through the Excel file and use the
histogram command to draw the histogram with eight bins.
2.22. A random variable X has the following PDF:
f
.
x
/
D
8
ˆ
ˆ
<
ˆ
ˆ
:
0 x < 40
0:025 40 x 80
0 x > 80:
Calculate its mean, standard deviation, and coefficient of variance.
2.23. e applied force X on a component can be treated as a random variable and described
by following PDF,
D
8
ˆ
<
ˆ
:
x
24
0 x 6
8 x
8
6 x 8:
(a) Determine the mean, standard devaluation of the random variable X.
(b) Determine the CDF.
(c) Calculate the probability of P .X > 1:5/.
2.24. e life of a device X (in thousands of hours) is represented by the PDF:
f
.
x
/
D 2:15e
2:15x
x 0:
(a) Determine the mean, standard devaluation of the random variable X.
(b) Determine the CDF.
(c) Calculate the probability of P .X > 2:5/.
2.25. A random variable is a realizing number of rolling a dice.
(a) Determine its PMF.
(b) Determine its CDF.
(c) Calculate the mean, standard deviation, and coefficient of variance of this random
variable.
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