Question 1

An experiment consists of tossing a coin, rolling a die, and observing the outcomes. Describe an appropriate sample space for this experiment.
A) {(H, H, H, H, H, H, T, T, T, T, T, T, 1, 2, 3, 4, 5, 6)}
B) {(1, 2, 3, 4, 5, 6), (H, T)}
C) {(H, 1), (T, 2), (H, 3), (T, 4), (H, 5), (T, 6), (T, 1), (T, 4), (T, 5), (H, 6)}
D) {(1, H, T), (2, H, T), (3, H, T), (4, H, T), (5, H, T), (6, H, T)}
E) {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}

Accepted Answer

The sample space should include all possible outcomes of the three events -- tossing a coin, rolling a die, and observing the outcomes. The sample space should consist of all possible ordered pairs of the form (coin outcome, die outcome). Therefore, option E is the best choice as it includes all possible outcomes of the experiment. Option A only includes the outcomes of the coin toss, while Option B only includes the outcomes of the coin toss and rolling the die. Option C includes only 10 outcomes, while the experiment has 12 possible outcomes. Option D includes outcomes that are not possible, such as (1, H, H).

Question 2

In a television game show, the winner is asked to select three prizes from five different prizes, A, B, C, D, and E. Describing a sample space of possible outcomes (order is not important) determine the number of points there are in the sample space corresponding to a selection that includes A.
A) 6
B) 9
C) 4
D) 2

Accepted Answer

6Since order is not important and the winner must select 3 prizes out of 5, and one of them must be A, we are left with choosing 2 more prizes out of the remaining 4 (B, C, D, E). This is a combination problem, calculated as 4 choose 2 (4C2), which equals 6.

Question 3

A sample of two transistors taken from a local electronics store was examined to determine whether the transistors were defective (d) or nondefective (n). What is an appropriate sample space for this experiment? ​
A) {dd, dn, nd}
B) {dd, dn, nd, nn}
C) {dd, nn, nd}
D) {(d,d), (n,n)}

Accepted Answer

The sample space should include all possible outcomes of the experiment, which in this case are the combinations of defective (d) and nondefective (n) transistors. The possible outcomes are: dd (both defective), dn (first defective, second nondefective), nd (first nondefective, second defective), and nn (both nondefective). Therefore, the appropriate sample space is {dd, dn, nd, nn}, which is only given in option B.

Question 4

Let be a sample space of an experiment and let be events of this experiment. Find the events F^c and E ∩ Gc. A) B) C) D)

Accepted Answer

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

Question 5

An experiment consists of selecting a card from a standard deck of playing cards and noting whether the card is black (B) or red (R). Describe an appropriate sample space for this experiment. ​
A) {BR, RB}
B) {R, R}
C) {R, B, B, R}
D) {B, R}
E) {B, B}

Accepted Answer

The sample space should include all possible outcomes, which in this case is either a black card or a red card. Therefore, the appropriate sample space is {B, R}.

Question 6

Let  be a sample space of an experiment with outcomes p, q and r. List all the events of this experiment. ​
A) Ø, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}, S
B) {p}, {q}, {r}
C) Ø, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}
D) {p}, {q}, {r}, {p, q}, {p, r}, {q, r}

Accepted Answer

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

Question 7

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.
A) {(1, 2), (2, 4), (3, 6)}
B) {(3, 1), (2, 6), (4, 4)}
C) {(1, 1), (2, 1), (1, 6)}
D) {(1, 2), (5, 3), (6, 6)}

Accepted Answer

The correct outcomes where the number on the second die is double the number on the first die are (1, 2), (2, 4), and (3, 6).

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 greater than the number that falls uppermost on the second die.
A) {(2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (3, 2), (4, 2), (5, 2), (6, 2), (4, 3), (5, 3), (6, 3), (5, 4), (6, 4), (6, 5)}
B) {(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (3, 2), (4, 2), (5, 2), (6, 2), (4, 3), (5, 3), (6, 3), (5, 4), (6, 4), (5, 6)}
C) {(2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (2, 3), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 5), (6, 4), (6, 5)}
D) Ø

Accepted Answer

In order for the number on the first die to be greater than the number on the second die, the first number in the ordered pair must be larger than the second number. We can see that this is true for all the outcomes in set A.

Question 9

Let S = {1, 2, 3, 4, 5, 6}, E = {1, 3, 5}, F = {2, 4, 6} and G = {2, 3}. Are the events F and G mutually exclusive? 
A) yes
B) no

Accepted Answer

Events F and G are not mutually exclusive because they share a common element, which is 2. Mutually exclusive events have no elements in common.

Question 10

Let  be a sample space of an experiment and let  , and  be events of this experiment. ​
Find the events E ∪ F and E ∩ F.
​
A) 
B) 
C) 
D)

Accepted Answer

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

Question 11

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 4 and the number falling uppermost on the other die is less than 4.
A) {(1, 4), (2, 4), (3, 4), (4, 3), (4, 2), (4, 1)}
B) {(4, 5), (4, 6), (5, 2), (6, 4)}
C) {(2, 3), (2, 4), (6, 5), (2, 6), (3, 2), (4, 2), (5, 1), (6, 2)}
D) Ø

Accepted Answer

The event that the number falling uppermost on one die is a 4 is {(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)}. The event that the number falling uppermost on the other die is less than 4 is {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}. The intersection of these two events is {(4, 1), (4, 2), (4, 3), (1, 4), (2, 4), (3, 4)}, which corresponds to choice A.

Question 12

An experiment consists of tossing a coin, rolling a die, and observing the outcomes. Describe the event "A head is tossed and an even number is rolled."
A) 
B) 
C) 
D)

Accepted Answer

The answer of An experiment consists of tossing a coin,...

Question 13

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 greater than or equal to 7.
A) {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
B) {(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 5), (4, 6), (5, 6)}
C) {(1, 6), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 3), (4, 5), (4, 6), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
D) Ø

Accepted Answer

The correct event includes all pairs where the sum of the two numbers is greater than or equal to 7. Choice C lists all combinations where this condition is met, covering all possibilities from the lowest (1,6) to the highest (6,6) that satisfy the condition.

Question 14

An experiment consists of selecting a card from a standard deck of playing cards and noting whether the card is black (B) or red (R). What are the events of this experiment? ​
A) {B, R}, {R, B}
B) {Ø}, {B}, {R}
C) {B}, {R}, {B, B}, {R, R}
D) {Ø}, {B}, {R}, {B, R}
E) {Ø}, {B, R}, {R, B}, {R, R}, {B, B}

Accepted Answer

The events are either selecting a black card (B), a red card (R), or selecting no card at all (Ø). Option D includes all of these events and no extra ones, making it the best choice. Option A includes two events that are the same except for order, which is unnecessary. Option B includes an empty set, which is not a valid event for this experiment. Option C includes events that involve selecting two cards, which is not how the experiment is described. Option E includes events that involve selecting both red and black cards, which is also not how the experiment is described.

Question 15

Let S = {1, 2, 3, 4, 5, 6}, E = {1, 3, 5}, F = {2, 4, 6} and G = {3, 4}. Are the events F and G complementary? 
A) no
B) yes

Accepted Answer

Two events are complementary if their union covers the entire sample space and their intersection is empty. However, F and G have an intersection of {4}, so they are not complementary.

Question 16

A die is cast and the number that falls uppermost is observed. Let E denote the event that the number shown is a 6 and let F denote the event that the number shown is an even number. ​
Are the events E and F mutually exclusive?
A) no
B) yes

Accepted Answer

Event E (rolling a 6) is a subset of event F (rolling an even number), since 6 is an even number. Therefore, they are not mutually exclusive.

Question 17

Let S = {1, 2, 3, 4, 5, 6}, E = {1, 3, 5}, F = {2, 4, 6} and G = {2, 3} . Find the event E ∪ F ∪ G.
A) E ∪ F ∪ G = {2, 3, 4, 5, 7}
B) E ∪ F ∪ G = {1, 3, 4, 5, 6}
C) E ∪ F ∪ G = {1, 2, 3, 4, 5, 6}
D) E ∪ F ∪ G = {1, 2, 3, 5, 6}

Accepted Answer

E ∪ F ∪ G means the union of sets E, F, and G. 
E = {1, 3, 5} 
F = {2, 4, 6} 
G = {2, 3}

The union of these sets contains all the elements present in at least one of the sets. Therefore, 
E ∪ F ∪ G = {1, 2, 3, 4, 5, 6}.

So, the answer is choice C.

Question 18

An experiment consists of selecting a letter at random from the letters in the word PENNSYLVANIA and observing the outcomes. What is an appropriate sample space for this experiment? ​
A) 
B) 
C) 
D)

Accepted Answer

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

Question 19

Let  be a sample space of an experiment and let  be events of this experiment. ​
Are the events E and F mutually exclusive?
A) yes
B) no

Accepted Answer

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

Question 20

Let S = {1, 2, 3, 4, 5, 6}, E = {2, 4, 6, }, F = {1, 3, 5} and G = {4, 5} . ​
Find the event (E ∩ F ∩ G)c.
​
A) (E ∩ F ∩ G)c = {1, 2, 3, 4, 5, 6}
B) (E ∩ F ∩ G)c = {1, 2, 3, 4, 5}
C) (E ∩ F ∩ G)c = {2, 3, 4, 5, 6}
D) (E ∩ F ∩ G)c = {1, 2, 5, 6, 9}

Accepted Answer

(E ∩ F ∩ G)c = {1, 2, 3, 4, 5, 6} because (E ∩ F ∩ G) = {4, 5} and its complement is the set of all elements in S that are not in {4, 5}.

Quiz 7: Probability

An experiment consists of tossing a coin, rolling a die, and observing the outcomes. Describe an appropriate sample space for this experiment.

In a television game show, the winner is asked to select three prizes from five different prizes, A, B, C, D, and E. Describing a sample space of possible outcomes (order is not important) determine the number of points there are in the sample space corresponding to a selection that includes A.

A sample of two transistors taken from a local electronics store was examined to determine whether the transistors were defective (d) or nondefective (n) . What is an appropriate sample space for this experiment?

Let be a sample space of an experiment and let be events of this experiment. Find the events F^c and E ∩ Gc.

An experiment consists of selecting a card from a standard deck of playing cards and noting whether the card is black (B) or red (R) . Describe an appropriate sample space for this experiment.

Let be a sample space of an experiment with outcomes p, q and r. List all the events of this experiment.

Let S = {1, 2, 3, 4, 5, 6}, E = {1, 3, 5}, F = {2, 4, 6} and G = {2, 3}. Are the events F and G mutually exclusive?

Let be a sample space of an experiment and let , and be events of this experiment. Find the events E ∪ F and E ∩ F.

An experiment consists of tossing a coin, rolling a die, and observing the outcomes. Describe the event "A head is tossed and an even number is rolled."

An experiment consists of selecting a card from a standard deck of playing cards and noting whether the card is black (B) or red (R) . What are the events of this experiment?

Let S = {1, 2, 3, 4, 5, 6}, E = {1, 3, 5}, F = {2, 4, 6} and G = {3, 4}. Are the events F and G complementary?

A die is cast and the number that falls uppermost is observed. Let E denote the event that the number shown is a 6 and let F denote the event that the number shown is an even number. Are the events E and F mutually exclusive?

Let S = {1, 2, 3, 4, 5, 6}, E = {1, 3, 5}, F = {2, 4, 6} and G = {2, 3} . Find the event E ∪ F ∪ G.

An experiment consists of selecting a letter at random from the letters in the word PENNSYLVANIA and observing the outcomes. What is an appropriate sample space for this experiment?

Let be a sample space of an experiment and let be events of this experiment. Are the events E and F mutually exclusive?

Let S = {1, 2, 3, 4, 5, 6}, E = {2, 4, 6, }, F = {1, 3, 5} and G = {4, 5} . Find the event (E ∩ F ∩ G) c.

Quiz 7: Probability

An experiment consists of tossing a coin, rolling a die, and observing the outcomes. Describe an appropriate sample space for this experiment.

In a television game show, the winner is asked to select three prizes from five different prizes, A, B, C, D, and E. Describing a sample space of possible outcomes (order is not important) determine the number of points there are in the sample space corresponding to a selection that includes A.

A sample of two transistors taken from a local electronics store was examined to determine whether the transistors were defective (d) or nondefective (n) . What is an appropriate sample space for this experiment? ​

Let be a sample space of an experiment and let be events of this experiment. ​ Find the events Fc and E ∩ Gc.

An experiment consists of selecting a card from a standard deck of playing cards and noting whether the card is black (B) or red (R) . Describe an appropriate sample space for this experiment. ​

Let be a sample space of an experiment with outcomes p, q and r. List all the events of this experiment. ​

Let S = {1, 2, 3, 4, 5, 6}, E = {1, 3, 5}, F = {2, 4, 6} and G = {2, 3}. Are the events F and G mutually exclusive?

Let be a sample space of an experiment and let , and be events of this experiment. ​ Find the events E ∪ F and E ∩ F. ​

An experiment consists of tossing a coin, rolling a die, and observing the outcomes. Describe the event "A head is tossed and an even number is rolled."

An experiment consists of selecting a card from a standard deck of playing cards and noting whether the card is black (B) or red (R) . What are the events of this experiment? ​

Let S = {1, 2, 3, 4, 5, 6}, E = {1, 3, 5}, F = {2, 4, 6} and G = {3, 4}. Are the events F and G complementary?

A die is cast and the number that falls uppermost is observed. Let E denote the event that the number shown is a 6 and let F denote the event that the number shown is an even number. ​ Are the events E and F mutually exclusive?

Let S = {1, 2, 3, 4, 5, 6}, E = {1, 3, 5}, F = {2, 4, 6} and G = {2, 3} . Find the event E ∪ F ∪ G.

An experiment consists of selecting a letter at random from the letters in the word PENNSYLVANIA and observing the outcomes. What is an appropriate sample space for this experiment? ​

Let be a sample space of an experiment and let be events of this experiment. ​ Are the events E and F mutually exclusive?

Let S = {1, 2, 3, 4, 5, 6}, E = {2, 4, 6, }, F = {1, 3, 5} and G = {4, 5} . ​ Find the event (E ∩ F ∩ G) c. ​

A sample of two transistors taken from a local electronics store was examined to determine whether the transistors were defective (d) or nondefective (n) . What is an appropriate sample space for this experiment?

Let be a sample space of an experiment and let be events of this experiment. Find the events F^c and E ∩ Gc.

An experiment consists of selecting a card from a standard deck of playing cards and noting whether the card is black (B) or red (R) . Describe an appropriate sample space for this experiment.

Let be a sample space of an experiment with outcomes p, q and r. List all the events of this experiment.

Let be a sample space of an experiment and let , and be events of this experiment. Find the events E ∪ F and E ∩ F.

An experiment consists of selecting a card from a standard deck of playing cards and noting whether the card is black (B) or red (R) . What are the events of this experiment?

A die is cast and the number that falls uppermost is observed. Let E denote the event that the number shown is a 6 and let F denote the event that the number shown is an even number. Are the events E and F mutually exclusive?

An experiment consists of selecting a letter at random from the letters in the word PENNSYLVANIA and observing the outcomes. What is an appropriate sample space for this experiment?

Let be a sample space of an experiment and let be events of this experiment. Are the events E and F mutually exclusive?

Let S = {1, 2, 3, 4, 5, 6}, E = {2, 4, 6, }, F = {1, 3, 5} and G = {4, 5} . Find the event (E ∩ F ∩ G) c.