Sample Spaces and the Algebra of Sets
Four dice are tossed: one red, one blue, one green, and one yellow. What outcomes make up the event A that the sum of the four faces showing equals 23?
A = {(5,6,6,6), (6,5,6,6), (6,6,5,6), (6,6,6,5)}
Sketch the regions in the xy-plane corresponding to A ∪ B and A ∩ B if A ={(x, y): 0 < x < 3, 0 < y < 3} and B ={(x, y): 2 < x < 4, 2 < y < 4}
During the university career fair, 5200 graduates were invited to attend two different career workshops. A total of 920 attended the first workshop, 780 attended the second workshop, while 3900 did not attend either workshop. How many graduates attended both workshops?
Let A and B denote the graduates who attended the first workshop and the second workshop, respectively. Then N(A) = 920, N(B) = 780, and N[(A ∪ B)^C] = 3900 (implying that N(A ∪ B) = 1300). Therefore, N(A ∩ B) = number who attended both workshops = 920 + 780 − 1300 = 400.
According to a consumer research organization, many online advertisements contain misleading information or excessive promotional claims. Among the advertisements they analyzed, 38% contained information they considered misleading, 24% made excessive promotional claims, and 9% contained both. What percentage of advertisements complied with the organization’s standards?
Let M and E denote the sets of advertisements with misleading information and excessive promotional claims, respectively. Then P(M) = 0.38, P(E) = 0.24, and P(M ∩ E) = 0.09. Therefore, P(ad complies) = P((M ∪ E)^C) = 1 − [P(M) + P(E) − P(M ∩ E)] = 0.47.
A university library is selecting two books from a collection of seven books for a special display. Three of the books are mathematics books, while the other four are from different subjects. What is the sample space associated with the experiment “The librarian selects two books”? Which outcomes are in the event A: At least one selected book is not a mathematics book?
Let m1, m2, and m3 denote the three mathematics books and b1, b2, b3, and b4, the four books from other subjects. Then
S = {(m1,m2), (m1,m3), (m2,m3), (m1,b1), (m1,b2), (m1,b3), (m1,b4), (m2,b1), (m2,b2), (m2,b3), (m2,b4), (m3,b1), (m3,b2), (m3,b3), (m3,b4), (b1,b2), (b1,b3), (b1,b4), (b2,b3), (b2,b4), (b3,b4)}.
The event A contains every outcome in S except (m1,m2), (m1,m3), and (m2,m3)
A medical monitoring system has six sensors divided into three groups. The two sensors within each group are wired in parallel; the three groups are wired in series. Let Aij denote the event “ith sensor in jth group fails,” i = 1, 2; j = 1, 2, 3. Let A be the event “System fails.” Write A in terms of the Aij’s.
The system fails if either the first group, the second group, or the third group fails. For any group to fail, though, both of its sensors must fail. Therefore,
A = (A11 ∩ A21) ∪ (A12 ∩ A22) ∪ (A13 ∩ A23).
Use Venn diagrams to suggest an equivalent way of representing the following events:
(a) (A ∩ B^C)^C
(b) B ∪(A ∪ B)^C
(c) A ∩(A ∩ B)^C
(a) A^C ∪ B
(b) A^C ∪ B
(c) A ∩ B^C
Suppose that three fair dice are tossed. Let Ai be the event that a 6 shows on the ith die, i = 1, 2, 3. Does P(A1 ∪ A2 ∪ A3) = 1/2 ? Explain.
No. P(A1 ∪ A2 ∪ A3) = P(at least one “6” appears) = 1 − P(no 6’s appear) = 1 − (5/6)^3 . The Ai’s are not mutually exclusive, so P(A1 ∪ A2 ∪ A3) ≠ P(A1) + P(A2) + P(A3).
Suppose a baseball player steps to the plate with the intention of trying to “coax” a base on balls by never swinging at a pitch. The umpire, of course, will necessarily call each pitch either a ball (B) or a strike (S). What outcomes make up the event A, that a batter walks on the sixth pitch? (Note: A batter “walks” if the fourth ball is called before the third strike.)
A = {SSBBBB, SBSBBB, SBBSBB, SBBBSB, BSSBBB, BSBSBB, BSBBSB, BBSSBB, BBSBSB, BBBSSB}
Suppose that three events—A, B, and C—are defined on a sample space S. Use the union, intersection, and complement operations to represent each of the following events:
(a) none of the three events occurs
(b) all three of the events occur
(c) only event A occurs
(a) A^C ∩ B^C ∩ C^C
(b) A ∩ B ∩ C
(c) A ∩ B^C ∩ C^C
For two events A and B defined on a sample space S, N(A ∩ B^C) = 18, N(A^C ∩ B) = 42, and N(A ∩ B) = 5. Given that N(S) = 100, how many outcomes belong to neither A nor B?
N[(A ∩ B)^C] = 100 − N(A ∪ B)
= 100 − [N(A^C ∩ B) + N(A ∩ B^C) + N(A ∩ B)]
= 100 − [42 + 18 + 5]
= 35
A cybersecurity team is testing four independent servers. Each server can either pass or fail a security scan, with both outcomes being equally likely. The team considers the system unstable if either exactly one server fails the scan or exactly one server passes the scan. What is the probability that the system will be classified as unstable after the first scan?
P(unstable) = 1 − P(no unstable result)
= 1 − P(all four pass or all four fail)
= 1 − 2/16
= 14/16 = 7/8
In a quality-control experiment, a factory inspects products one at a time. Each product is assigned one of five possible grades: A, B, C, D, or F. If the first product receives grade A or B, the inspection process ends and the batch is accepted. If the first product receives grade C, D, or F, that grade becomes the “reference grade.” The factory continues inspecting products until either the same reference grade appears again, in which case the batch is accepted, or grade F appears, in which case the batch is rejected. Characterize the sample outcomes contained in the event “The batch is accepted with a reference grade of D.”
(D, D), (D, not F or not D, D), (D, not F or not D, not F or not D, D), …
Suppose that each of the eleven letters in the word MATHEMATICS is written on a chip. Define the events F, V, and N as follows:
F: letters that occur in both “MATHEMATICS” and “CALCULUS”
V: letters that occur exactly twice in “MATHEMATICS”
N: letters that come after M in the alphabet
Which chips make up the following events?
(a) F ∩ V ∩ N
(b) F^C ∩ V ∩ N^C
(c) F ∩ V^C ∩ N
(a) {∅} (b) {M1,M2} (c) {S}
A total of 1,500 students at a university enrolled in either Computer Science or Mathematics courses during the academic year. Of these students, 900 completed a programming course, and 650 completed a statistics course. Moreover, 350 students completed both programming and statistics courses. What proportion of the 1,500 students completed neither a programming nor a statistics course?
Let A be the set of students who completed a programming course and B be the set of students who completed a statistics course. We are given that N(A) = 900, N(B) = 650, and N(A ∩ B) = 350. Then N(A^C ∩ B^C) = N[(A ∪ B)^C] = 1500 − N(A ∪ B)
= 1500 − [N(A) + N(B) − N(A ∩ B)]
= 1500 − [900 + 650 − 350]
= 300.
300/1500 = 1/5
A box contains thirty cards numbered 1 through 30. One card is selected at random. Let A be the event that the number is divisible by 3 and let B be the event that the number is divisible by 5. Find P(A ∪ B).
A = {3, 6, 9, …, 30} → 10 numbers
B = {5, 10, 15, …, 30} → 6 numbers
A ∩ B = {15, 30} → 2 numbers
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
= 10/30 + 6/30 − 2/30
= 14/30 = 7/15.