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 twelve letters in the word TESSELLATION is written on a chip. Define the events F, R, and C as follows:
F: letters in first half of alphabet
R: letters that are repeated
V: letters that are vowels
Which chips make up the following events?
(a) F ∩ R ∩ V
(b) F^C ∩ R ∩ V^C
(c) F ∩ R^C ∩ V
(a) {E1, E2} (b) {S1, S2, T1, T2} (c) {A, I}
A total of twelve hundred graduates of State Tech have gotten into medical school in the past several years. Of that number, one thousand earned scores of twenty-seven or higher on the MCAT and four hundred had GPAs that were 3.5 or higher. Moreover, three hundred had MCATs that were twenty-seven or higher and GPAs that were 3.5 or higher. What proportion of those twelve hundred graduates got into medical school with an MCAT lower than twenty-seven and a GPA below 3.5?
Let A be the set of those with MCAT scores ≥ 27 and B be the set of those with GPAs ≥ 3.5. We are given that N(a) = 1000, N(b) = 400, and N(A ∩ B) = 300. Then N(A^C ∩ B^C) = N[(A ∪ B)^C] = 1200 − N(A ∪ B) = 1200 − [(N(a) + N(b) − N(A ∩ B)] = 1200 − [(1000 + 400 − 300] = 100. The requested proportion is 100/1200.
An urn contains twenty-four chips, numbered 1 through 24. One is drawn at random. Let A be the event that the number is divisible by 2 and let B be the event that the number is divisible by 3. Find P(A ∪ B).
A = {2, 4, 6, …, 24}; B = {3, 6, 9, …, 24); A ∩ B = {6, 12, 18, 24}. Therefore, P(A ∪ B) = P(a) + P(b) − P(A ∩ B) = 12/24+8/24-4/24=16/24
A probability-minded despot offers a convicted murderer a final chance to gain his release. The prisoner is given twenty chips, ten white and ten black. All twenty are to be placed into two urns, according to any allocation scheme the prisoner wishes, with the one proviso being that each urn contain at least one chip. The executioner will then pick one of the two urns at random and from that urn, one chip at random. If the chip selected is white, the prisoner will be set free; if it is black, he “buys the farm.” Characterize the sample space describing the prisoner’s possible allocation options. (Intuitively, which allocation affords the prisoner the greatest chance of survival?)
Let (x, y) denote the strategy of putting x white chips and y black chips in the first urn (which results in 10 − x white chips and 10 − y black chips being in the second urn). Then S = {(x,y) : x=0,1,...,10, y=0,1,...,10, and 1≤ x+y ≤19 } . Intuitively, the optimal strategies are (1, 0) and (9, 10).
A coin is tossed four times and the resulting sequence of heads and/or tails is recorded. Define the events A, B, and C as follows:
A: exactly two heads appear
B: heads and tails alternate
C: first two tosses are heads
(a) Which events, if any, are mutually exclusive? (b) Which events, if any, are subsets of other sets?
(a) B and C (b) B is a subset of A
A poll conducted by a potential presidential candidate asked two questions: (1) Do you support the candidate’s position on taxes? and (2) Do you support the candidate’s position on homeland security? A total of twelve hundred responses were received; six hundred said “yes” to the first question and four hundred said “yes” to the second. If three hundred respondents said “no” to the taxes question and “yes” to the homeland security question, how many said “yes” to the taxes question but “no” to the homeland security question?
Let A be the set of those saying “yes” to the first question and B be the set of those saying “yes” to the second question. We are given that N(a) = 600, N(b) = 400, and N(A^C ∩ B) = 300. Then N(A ∩ B) = N(b) − N(A^C ∩ B) = 400 − 300 = 100. N(A ∩ B^C) = N(a) − N(A ∩ B) = 600 − 100 = 50
Consolidated Industries has come under considerable pressure to eliminate its seemingly discriminatory hiring practices. Company officials have agreed that during the next five years, 60% of their new employees will be females and 30% will be minorities. One out of four new employees, though, will be a white male. What percentage of their new hires will be minority females?
Let F: female is hired and T: minority is hired. Then P(f) = 0.60, P(T) = 0.30, and P(F^C ∩ T^C ) = 0.25 = 1 − P(F ∪ T). Since P(F ∪ T) = 0.75, P(F ∩ T) = 0.60 + 0.30 − 0.75 = 0.15