Question 1

Let S be a sample space for an experiment and let G and A be events of this experiment. Show that the events  and  are mutually exclusive. ​
Hint: Use De Morgan's law.
​
A) By De Morgan's law,  , so the events are mutually exclusive.
B) By De Morgan's law,  , so the events are mutually exclusive.
C) By De Morgan's law,  , so the events are mutually exclusive.
D) By De Morgan's law,  , so the events are mutually exclusive.
E) By De Morgan's law,  , so the events are not mutually exclusive.

Accepted Answer

By De Morgan's law, the complement of G union A is equal to the complement of G intersect complement of A. Therefore, if the events G and A are mutually exclusive (i.e. they have no common outcomes), then their complements G^c and A^c must have outcomes in common. This means that G^c intersect A^c is non-empty. In this case, the complement of event 11eab448_e7c6_ed33_acdb_530a89fa849e_TB6027_11 is equal to event 11eab448_e7c6_ed35_acdb_3bd83e257e4c_TB6027_11, and the complement of event 11eab448_e7c6_ed34_acdb_db3febbaf014_TB6027_11 is equal to event 11eab448_e7c7_1446_acdb_1f606ead45c9_TB6027_11. Since these two events are complementary, they have no outcomes in common, and therefore the original events 11eab448_e7c6_ed33_acdb_530a89fa849e_TB6027_11 and 11eab448_e7c6_ed34_acdb_db3febbaf014_TB6027_11 are mutually exclusive.

Question 2

Let S be any sample space and N, F, and E be any three events associated with the experiment. Describe the following event, using the symbols ∪, ∩, and ^c. The event that E does not occur. A) E^c B) C) D) E)

Accepted Answer

The answer of Let S be any sample space and...

Question 3

A certain airport hotel operates a shuttle bus service between the hotel and the airport. The maximum capacity of a bus is 20 passengers. On alternate trips of the shuttle bus over a period of 1 wk, the hotel manager kept a record of the number of passengers arriving at the hotel in each bus. Describe the event E that a shuttle bus carried fewer than twelve passengers.
A) {0, 1, 2, 3, 4, 5, 6, 7, 8}
B) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
C) {0, 1, 2, 3, 4, 5, 6, 7}
D) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

Accepted Answer

The event E is defined as the shuttle bus carrying fewer than 12 passengers. Since the maximum capacity of the bus is 20 passengers, any number of passengers between 0 and 11 (inclusive) would satisfy this event. Therefore, the set of possible outcomes for event E is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. The correct answer is D.

Question 4

An experiment consists of casting a pair of dice and observing the number that falls uppermost on each die. We may represent each outcome of the experiment by an ordered pair of numbers, the first representing the number that appears uppermost on the first die and the second representing the number that appears uppermost on the second die. Consider the sample space
S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
Determine the event that the sum of the numbers falling uppermost is less than or equal to 6.
Express your answer using set notation.

Accepted Answer

The answer of An experiment consists of casting a pair...

Question 5

As part of a quality-control procedure, an inspector at Bristol Farms randomly selects ten eggs from each consignment of eggs he receives and records the number of broken eggs. Describe the event F that at least nine eggs are broken.
A) {9, 10}
B) {7, 8, 9, 10}
C) {6, 7, 8, 9, 10}
D) {0, 1, 2, 3, 4, 5}

Accepted Answer

The event F is that at least nine eggs are broken. This means that either nine or ten eggs are broken, which corresponds to choice A. The other choices include smaller numbers of broken eggs, which are not included in the event F.

Question 6

Eight players, A, B, C, D, E, F, G, and H, are competing in a series of elimination matches of a tennis tournament in which the winner of each preliminary match will advance to the semifinals and the winner of the semifinals will advance to the finals. An outline of the scheduled matches follows. Describe a sample space listing the possible participants in the finals. 
A) S = {(A,D), (A,F), (A,H), (C,B), (C,F), (C,H), (E,B), (E,D), (E, H), (G,B),
(G,D), (G,F)}
B) S = {(A,B), (A,D), (A,F), (A,H), (C,B), (C,D), (C,F), (C,H), (E,B), (E,D),
(E,F), (E, H), (G,B), (G,D), (G,F), (G,H)}
C) S = {(A,E), (A,F), (A,G), (A,H), (B,E), (B,F), (B,G), (B,H), (C, E), (C,F),
(C,G), (C,H), (D,E), (D,F), (D,G), (D,H)}
D) S = {(A,B), (A,C), (A,D), (B,С), (B,D), (C,D), (E,F), (E,G), (E, H), (F,G),
(F,H), (G,H)}
E) None of the above.

Accepted Answer

The answer of Eight players, A, B, C, D, E,...

Question 7

The customer service department of Universal Instruments, manufacturer of the Galaxy home computer, conducted a survey among customers who had returned their purchase registration cards. Purchasers of its deluxe model home computer were asked to report the length of time (t) in days before service was required. Describe a sample space corresponding to this survey.
A) {t | t > 0}
B) {t | 0 < t ≤ 370}
C) {t | 100 ≤ t ≤ 370}
D) {t | 0 < t ≤ 100}

Accepted Answer

The sample space should include all possible values of t (time in days before service was required) greater than 0, as there is no specified upper limit in the survey description.

Question 8

An experiment consists of casting a pair of dice and observing the number that falls uppermost on each die. We may represent each outcome of the experiment by an ordered pair of numbers, the first representing the number that appears uppermost on the first die and the second representing the number that appears uppermost on the second die. Consider the sample space
S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
Determine the event that the number that falls uppermost on the first die is less than the number that falls uppermost on the second die.
Express your answer using set notation.

Accepted Answer

The answer of An experiment consists of casting a pair...

Question 9

An experiment consists of casting a pair of dice and observing the number that falls uppermost on each die. We may represent each outcome of the experiment by an ordered pair of numbers, the first representing the number that appears uppermost on the first die and the second representing the number that appears uppermost on the second die. Consider the sample space
S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
Determine the event that the number that falls uppermost on the second die is double that of the number that falls on the first die.
Express your answer using set notation.

Accepted Answer

The answer of An experiment consists of casting a pair...

Question 10

Let  be a sample space of an experiment with outcomes k, l and m. List all the events of this experiment.
​
Express your answer using set notation.

Accepted Answer

The answer of Let  be a sample space of...

Question 11

Let  ,  ,  and  .
Find the event E ∪ F ∪ G.
Express your answer using set notation.

Accepted Answer

The answer of Let  ,  ,  and...

Question 12

Robin purchased shares of a machine tool company and shares of an airline company. Let E be the event that the shares of the machine tool company increase in value over the next 5 months, and let F be the event that the shares of the airline company increase in value over the next 5 months. Using the symbols ∪, ∩ and ^c describe the event that the shares of only one of the two companies increase in value. A) (E ∩ F^c) ∪ (E^c ∩ F) B) E ∪ F C) E^c ∩ F^c D) Ø

Accepted Answer

The event that the shares of only one of the two companies increase in value can occur in one of the two ways: either the shares of the machine tool company increase in value, while the shares of the airline company do not increase in value, or the shares of the airline company increase in value, while the shares of the machine tool company do not increase in value. Therefore, this event can be described as the union of two disjoint events: (E ∩ F' ) and (E' ∩ F). Using set notation, this can be expressed as (E ∩ F^c) ∪ (E^c ∩ F).

Question 13

Human blood is classified by the presence or absence of three main antigens (A, B, and Rh). When a blood specimen is typed, the presence of the A and/or B antigen is indicated by listing the letter A and/or the letter B. If neither the A nor B antigen is present, the letter O is used. The presence or absence of the Rh antigen is indicated by the symbols + or -, respectively. Thus, if a blood specimen is classified as AB⁺, it contains the A and the B antigens as well as the Rh antigen. Similarly, O^- blood contains none of the three antigens. Using this information, determine the sample space corresponding to the different blood groups. A) {AB⁺, AB^-, AO⁺, BO⁺, AO^-, BO^-, O⁺, O^-} B) {AB⁺, AB^-, O⁺, O^-} C) {A⁺, B⁺, A^-, B^-, O⁺, O^-} D) {AB⁺, AB^-, A⁺, B⁺, A^-, B^-, O⁺, O^-, ABO^-, AO⁺, AO^-, BO⁺, BO^-} E) {AB⁺, AB^-, A⁺, B⁺, A^-, B^-, O⁺, O^-}

Accepted Answer

The sample space for blood groups based on the presence or absence of A, B, and Rh antigens includes all combinations of these antigens being present or absent. This includes blood groups with A, B, both (AB), or neither (O) antigen, and each of these can be Rh positive (+) or Rh negative (-), leading to eight possible combinations.

Question 14

Let S be any sample space and K, E, and M be any three events associated with the experiment. Describe the following event, using the symbols ∪, ∩, and ^c. The event that M does not occur.

Accepted Answer

The answer of Let S be any sample space and...

Question 15

An experiment consists of casting a pair of dice and observing the number that falls uppermost on each die. We may represent each outcome of the experiment by an ordered pair of numbers, the first representing the number that appears uppermost on the first die and the second representing the number that appears uppermost on the second die. Consider the sample space
S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
Determine the event that the number falling uppermost on one die is a 2 and the number falling uppermost on the other die is greater than 2.
Express your answer using set notation.

Accepted Answer

The answer of An experiment consists of casting a pair...

Question 16

Let  ,  ,  and  .
​
Find the event  .
​
Express your answer using set notation.

Accepted Answer

The answer of Let  ,  ,  and...

Question 17

An experiment consists of selecting a letter at random from the letters in the word LOUISIANA and observing the outcomes. What is an appropriate sample space for this experiment?
​
Express your answer using set notation.

Accepted Answer

The answer of An experiment consists of selecting a letter...

Question 18

An opinion poll is conducted among a state's electorate to determine the relationship between their income levels and their stands on a proposition aimed at reducing state income taxes. Voters are classified as belonging to either the low-, middle-, or upper-income group. They are asked whether they favor, oppose, or are undecided about the proposition. Let the letters L, M, and U represent the low-, middle-, and upper-income groups, respectively, and let the letters f, o, and u represent the responses - favor, oppose, and undecided, respectively. ​
Describe the event E that a respondent does not oppose the proposition and does not belong to the upper-income group.
​
A) E = {(L,o), (L,u), (L,f), (M,o), (M,u), (M,f), (U,o), (U,u)}
B) E = {(L,f), (L,u), (M,f), (M,u)}
C) E = {(L,f), (M,f), (U,o), (U,u)}
D) E = {(L,o), (L,u), (L,f), (M,o), (M,u), (M,f)}
E) E = {(L,o), (L,u), (M,o), (M,u), (U,f)}

Accepted Answer

Event E includes respondents who do not oppose the proposition (thus excluding any pair with 'o') and do not belong to the upper-income group (thus excluding any pair with 'U'), leaving only those in the low- and middle-income groups who either favor or are undecided about the proposition.

Question 19

Let  be a sample space associated with an experiment. How many subsets of S contain either the number 2 or the number 1?
A) 5
B) 7
C) 4
D) 6
E) 3

Accepted Answer

To find the number of subsets that contain either 2 or 1, we can first find the total number of subsets and subtract the number of subsets that do not contain 1 or 2.

The total number of subsets of a set with n elements is 2^n. In this case, there are 11 elements, so the total number of subsets is 2^11 = 2048.

The number of subsets that do not contain 1 or 2 can be found by considering the remaining 9 elements. Each of these can either be in the subset or not, giving 2 options for each element and a total of 2^9 = 512 possible subsets.

Therefore, the number of subsets that contain either 1 or 2 is 2048 - 512 = 1536.

To check our work, we can also count the number of subsets that contain both 1 and 2. There are 9 remaining elements, and we need to choose subsets from these, so the number of subsets that contain both 1 and 2 is 2^9 - 2 = 510.

The number of subsets that contain either 1 or 2 is therefore 1536 + 510 = 2046.

Therefore, the answer is D) 6.

Question 20

A time study was conducted by the production manager of Vista Vision to determine the length of time (t) in minutes required by an assembly worker to complete a certain task during the assembly of its Pulsar color television sets. Describe a sample space corresponding to this time study.
A) {t | 0 < t < 8}
B) {t | 8 ≤ t ≤ 10}
C) {t | 0< t ≤ 10}
D) {t | t > 0}

Accepted Answer

The answer of A time study was conducted by the...

Quiz 7: Probability

Let S be a sample space for an experiment and let G and A be events of this experiment. Show that the events and are mutually exclusive. Hint: Use De Morgan's law.

Let S be any sample space and N, F, and E be any three events associated with the experiment. Describe the following event, using the symbols ∪, ∩, and ^c. The event that E does not occur.

As part of a quality-control procedure, an inspector at Bristol Farms randomly selects ten eggs from each consignment of eggs he receives and records the number of broken eggs. Describe the event F that at least nine eggs are broken.

Let be a sample space of an experiment with outcomes k, l and m. List all the events of this experiment. Express your answer using set notation.

Let , , and . Find the event E ∪ F ∪ G. Express your answer using set notation.

Let S be any sample space and K, E, and M be any three events associated with the experiment. Describe the following event, using the symbols ∪, ∩, and ^c. The event that M does not occur.

Let , , and . Find the event . Express your answer using set notation.

An experiment consists of selecting a letter at random from the letters in the word LOUISIANA and observing the outcomes. What is an appropriate sample space for this experiment? Express your answer using set notation.

Let be a sample space associated with an experiment. How many subsets of S contain either the number 2 or the number 1?

A time study was conducted by the production manager of Vista Vision to determine the length of time (t) in minutes required by an assembly worker to complete a certain task during the assembly of its Pulsar color television sets. Describe a sample space corresponding to this time study.

Quiz 7: Probability

Let S be a sample space for an experiment and let G and A be events of this experiment. Show that the events and are mutually exclusive. ​ Hint: Use De Morgan's law. ​

Let S be any sample space and N, F, and E be any three events associated with the experiment. Describe the following event, using the symbols ∪, ∩, and c. ​ The event that E does not occur. ​

As part of a quality-control procedure, an inspector at Bristol Farms randomly selects ten eggs from each consignment of eggs he receives and records the number of broken eggs. Describe the event F that at least nine eggs are broken.

Let be a sample space of an experiment with outcomes k, l and m. List all the events of this experiment. ​ Express your answer using set notation.

Let , , and . Find the event E ∪ F ∪ G. Express your answer using set notation.

Let S be any sample space and K, E, and M be any three events associated with the experiment. Describe the following event, using the symbols ∪, ∩, and c. ​ The event that M does not occur.

Let , , and . ​ Find the event . ​ Express your answer using set notation.

An experiment consists of selecting a letter at random from the letters in the word LOUISIANA and observing the outcomes. What is an appropriate sample space for this experiment? ​ Express your answer using set notation.

Let be a sample space associated with an experiment. How many subsets of S contain either the number 2 or the number 1?

A time study was conducted by the production manager of Vista Vision to determine the length of time (t) in minutes required by an assembly worker to complete a certain task during the assembly of its Pulsar color television sets. Describe a sample space corresponding to this time study.

Let S be a sample space for an experiment and let G and A be events of this experiment. Show that the events and are mutually exclusive. Hint: Use De Morgan's law.

Let S be any sample space and N, F, and E be any three events associated with the experiment. Describe the following event, using the symbols ∪, ∩, and ^c. The event that E does not occur.

Let be a sample space of an experiment with outcomes k, l and m. List all the events of this experiment. Express your answer using set notation.

Let S be any sample space and K, E, and M be any three events associated with the experiment. Describe the following event, using the symbols ∪, ∩, and ^c. The event that M does not occur.

Let , , and . Find the event . Express your answer using set notation.

An experiment consists of selecting a letter at random from the letters in the word LOUISIANA and observing the outcomes. What is an appropriate sample space for this experiment? Express your answer using set notation.