ISE 130 (Fall 2025)
ISE 130 - Engineering Probability and Statistics
Homework 1
Due: Sep 06, 2025 @ 11:59 PM
Show your work in full detail to maximize partial credit for an incorrect answer.
Problem 1 (10 points):
In an injection-molding operation, the length and width, denoted as X and Y, respectively, of
each molded part are evaluated. Let
A denote the event of 42 < X < 52 centimeters
B denote the event of 11 < Y < 21 centimeters
Construct a Venn diagram that includes these events. Shade the areas that represent the
following: (a) A; (b) A B; (c) A B; (d) A B; (e) If these events were mutually exclusive,
how successful would this production operation be? Would the process produce parts with X =
50 centimeters and Y = 15 centimeters?
Solutions:
B B
21 21
11 11
A A
42 52 42 52
(a) A (b) A B
B B
21 21
11 11
A A
42 52 42 52
(c) A B (d) A B
(e) If the events are mutually exclusive, then AB is null set. Therefore, the process does not
produce product parts with X = 50 cm and Y = 15 cm. The process would not be successful.
1
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, ISE 130 (Fall 2025)
Problem 2 (6 points):
A sample of two items is selected without replacement from a batch. Describe the (ordered)
sample space for each of the following batches:
(a) The batch contains the items {a, b, c, d}.
(b) The batch contains 4 defective items and 20 good items.
(c) The batch contains 1 defective item and 20 good items.
Solutions:
(a) {ab, ac, ad, bc, bd, cd, ba, ca, da, cb, db, dc}
(b) Let d and g denote defective and good, respectively. Then S = {gg, gd, dg, dd}
(c) S = {gd, dg, gg}
Problem 3 (9 points):
A batch of 140 semiconductor chips is inspected by choosing a sample of 5 chips. Assume 10 of
the chips do not conform to customer requirements.
(a) How many different samples are possible?
(b) How many samples of five contain exactly one nonconforming chip?
(c) How many samples of five contain at least one nonconforming chip?
Solutions:
140!
(a) The number of samples of size five is (140
5
) = 5!135! = 416,965,528
130!
(b) There are 10 ways of selecting one nonconforming chip and there are (130 4
) = 4!126! =
11,358,880 ways of selecting four conforming chips. Therefore, the number of samples that
contain exactly one nonconforming chip is 10× (130
4
) = 113,588,800
(c) The number of samples that contain at least one nonconforming chip is the total number of
samples (140
5
) minus the number of samples that contain no nonconforming chips (130
5
). That
140! 130!
is (140
5
) − (130
5
) = 5!135! − 5!125! = 130,721,752
Problem 4 (6 points):
In the layout of a printed circuit board for an electronic product, 12 different locations can
accommodate chips. Five chips are to be placed on the circuit board.
(a) If the five chips that are placed on the board are of the same type, how many different layouts
are possible?
(a) If five different types of chips are to be placed on the board, how many different layouts are
possible?
Solutions:
(a) If the chips are of the same type, then every subset of 5 locations chosen from the 12 results
12!
in a different layout. Therefore, (125
) = 5!7! = 792 layouts are possible.
(b) If the chips are of different types, then every arrangement of 5 locations selected from the
12!
12 results in a different layout. Therefore, 𝑃512 = 7! = 95,040 layouts are possible.
2
This study source was downloaded by 100000822442696 from CourseHero.com on 12-03-2025 08:09:39 GMT -06:00
https://www.coursehero.com/file/251905339/HW1-Solutionspdf/
ISE 130 - Engineering Probability and Statistics
Homework 1
Due: Sep 06, 2025 @ 11:59 PM
Show your work in full detail to maximize partial credit for an incorrect answer.
Problem 1 (10 points):
In an injection-molding operation, the length and width, denoted as X and Y, respectively, of
each molded part are evaluated. Let
A denote the event of 42 < X < 52 centimeters
B denote the event of 11 < Y < 21 centimeters
Construct a Venn diagram that includes these events. Shade the areas that represent the
following: (a) A; (b) A B; (c) A B; (d) A B; (e) If these events were mutually exclusive,
how successful would this production operation be? Would the process produce parts with X =
50 centimeters and Y = 15 centimeters?
Solutions:
B B
21 21
11 11
A A
42 52 42 52
(a) A (b) A B
B B
21 21
11 11
A A
42 52 42 52
(c) A B (d) A B
(e) If the events are mutually exclusive, then AB is null set. Therefore, the process does not
produce product parts with X = 50 cm and Y = 15 cm. The process would not be successful.
1
This study source was downloaded by 100000822442696 from CourseHero.com on 12-03-2025 08:09:39 GMT -06:00
https://www.coursehero.com/file/251905339/HW1-Solutionspdf/
, ISE 130 (Fall 2025)
Problem 2 (6 points):
A sample of two items is selected without replacement from a batch. Describe the (ordered)
sample space for each of the following batches:
(a) The batch contains the items {a, b, c, d}.
(b) The batch contains 4 defective items and 20 good items.
(c) The batch contains 1 defective item and 20 good items.
Solutions:
(a) {ab, ac, ad, bc, bd, cd, ba, ca, da, cb, db, dc}
(b) Let d and g denote defective and good, respectively. Then S = {gg, gd, dg, dd}
(c) S = {gd, dg, gg}
Problem 3 (9 points):
A batch of 140 semiconductor chips is inspected by choosing a sample of 5 chips. Assume 10 of
the chips do not conform to customer requirements.
(a) How many different samples are possible?
(b) How many samples of five contain exactly one nonconforming chip?
(c) How many samples of five contain at least one nonconforming chip?
Solutions:
140!
(a) The number of samples of size five is (140
5
) = 5!135! = 416,965,528
130!
(b) There are 10 ways of selecting one nonconforming chip and there are (130 4
) = 4!126! =
11,358,880 ways of selecting four conforming chips. Therefore, the number of samples that
contain exactly one nonconforming chip is 10× (130
4
) = 113,588,800
(c) The number of samples that contain at least one nonconforming chip is the total number of
samples (140
5
) minus the number of samples that contain no nonconforming chips (130
5
). That
140! 130!
is (140
5
) − (130
5
) = 5!135! − 5!125! = 130,721,752
Problem 4 (6 points):
In the layout of a printed circuit board for an electronic product, 12 different locations can
accommodate chips. Five chips are to be placed on the circuit board.
(a) If the five chips that are placed on the board are of the same type, how many different layouts
are possible?
(a) If five different types of chips are to be placed on the board, how many different layouts are
possible?
Solutions:
(a) If the chips are of the same type, then every subset of 5 locations chosen from the 12 results
12!
in a different layout. Therefore, (125
) = 5!7! = 792 layouts are possible.
(b) If the chips are of different types, then every arrangement of 5 locations selected from the
12!
12 results in a different layout. Therefore, 𝑃512 = 7! = 95,040 layouts are possible.
2
This study source was downloaded by 100000822442696 from CourseHero.com on 12-03-2025 08:09:39 GMT -06:00
https://www.coursehero.com/file/251905339/HW1-Solutionspdf/
