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WorksheetsChapter 3 Probability Notes for Final Practice
Total questions: 190
Worksheet time: 2hrs 35mins
Which of the following is used to find the number of ways two or more events can occur in probability?
Fundamental Counting Principle
Law of Large Numbers
Addition Rule
Multiplication Table
What is the term for the set of all possible outcomes in a probability experiment?
Sample space
Event space
Probability set
Outcome list
Which type of probability is based on actual experiments or observations?
Empirical probability
Classical probability
Subjective probability
Theoretical probability
What is the probability of the complement of an event?
1 minus the probability of the event
The same as the probability of the event
Always zero
Always one
What is a probability experiment?
An action, or trial, through which specific results (counts, measurements, or responses) are obtained.
The set of all possible outcomes of a probability experiment.
The result of a single trial in a probability experiment.
A subset of the sample space consisting of one or more outcomes.
What does a tree diagram help determine in a sample space?
The number of possible outcomes
The average blood type
The fastest survey method
The most common blood type
What is a simple event in probability?
An event that consists of a single outcome.
An event that consists of multiple outcomes.
An event that cannot occur.
An event that is impossible.
Which of the following is an example of a simple event?
Tossing heads and rolling a 3.
Tossing heads and rolling an even number.
Tossing tails and rolling a 2 or 4.
Tossing heads or tails and rolling any number.
If an event consists of more than one outcome, what is it called?
Not a simple event.
A simple event.
A certain event.
An impossible event.
Which set represents the event "Tossing heads and rolling an even number"?
{H2, H4, H6}
{H3}
{T1, T2, T3}
{H1, H3, H5}
What makes Event A in the example a simple event?
It has only one outcome: choosing the specific defective machine part.
It involves multiple outcomes.
It requires selecting more than one machine part.
It is not related to quality assurance.
If you roll a six-sided die, what are the possible outcomes for the event "rolling at least a 4"?
Rolling a 4, a 5, or a 6
Rolling a 1, a 2, or a 3
Rolling only a 4
Rolling only a 6
Why is the event "rolling at least a 4" on a six-sided die not considered a simple event?
Because it has more than one outcome
Because it has only one outcome
Because it is impossible to roll a 4
Because a die has more than six sides
What does the Fundamental Counting Principle state about the number of ways two events can occur in sequence?
The number of ways is the sum of the ways each event can occur.
The number of ways is the product of the ways each event can occur.
The number of ways is the difference between the ways each event can occur.
The number of ways is the quotient of the ways each event can occur.
If one event can occur in m ways and a second event can occur in n ways, how many ways can the two events occur in sequence according to the Fundamental Counting Principle?
m + n
m - n
m × n
m ÷ n
Can the Fundamental Counting Principle be extended to more than two events occurring in sequence?
Yes, it can be extended to any number of events.
No, it only applies to two events.
Yes, but only up to three events.
No, it applies only to independent events.
Which principle is being used in the example to determine the number of ways to select a car manufacturer, car size, and color?
Fundamental Counting Principle
Principle of Least Action
Principle of Superposition
Principle of Relativity
If you can choose from 3 manufacturers, 2 car sizes, and 4 colors, how many different ways can you select one manufacturer, one car size, and one color?
12
18
24
9
If an access code for a car’s security system consists of four digits, each digit from 0 through 9, how many access codes are possible if each digit can be used only once and not repeated?
5040
10000
3024
9999
If an access code for a car’s security system consists of four digits, each digit from 0 through 9, how many access codes are possible if each digit can be repeated?
10000
5040
3024
9000
If an access code for a car’s security system consists of four digits, each digit from 0 through 9, how many access codes are possible if each digit can be repeated but the first digit cannot be 0 or 1?
8000
10000
5040
9000
What is a key characteristic of Classical (theoretical) Probability?
Each outcome in a sample space is equally likely.
Outcomes are based on past experiments.
Probability is determined by intuition.
Only one outcome is possible.
Which formula represents the probability of an event E in classical probability?
P(E) = Number of outcomes in event E / Number of outcomes in sample space
P(E) = Number of outcomes in sample space / Number of outcomes in event E
P(E) = Number of outcomes in event E × Number of outcomes in sample space
P(E) = Number of outcomes in sample space - Number of outcomes in event E
What is empirical (statistical) probability based on?
Observations obtained from probability experiments
Theoretical calculations only
Guesswork and assumptions
Historical data unrelated to experiments
Empirical probability is also known as:
Theoretical probability
Statistical probability
Classical probability
Subjective probability
Which formula represents the empirical probability of an event E?
P(E) = f / n
P(E) = n / f
P(E) = f × n
P(E) = f + n
In the formula P(E) = f / n, what does 'n' represent?
Frequency of event E
Total frequency
Probability of event E
Number of possible outcomes
What does the notation n = Σf mean in the context of empirical probability?
n is the sum of all frequencies
n is the product of all frequencies
n is the difference between frequencies
n is the average of all frequencies
Which type of probability is based on intuition, educated guesses, and estimates?
Subjective Probability
Experimental Probability
Theoretical Probability
Classical Probability
Which of the following is an example of subjective probability?
A doctor feels a patient has a 90% chance of full recovery.
Flipping a coin and getting heads.
Rolling a die and getting a 6.
Calculating the probability using a formula.
What does the Law of Large Numbers state?
As an experiment is repeated, the empirical probability of an event approaches the theoretical probability.
As an experiment is repeated, the theoretical probability of an event becomes less accurate.
The probability of an event remains constant regardless of the number of trials.
The empirical probability of an event decreases as the experiment is repeated.
According to the Law of Large Numbers, what happens when you toss a coin many times?
The proportion of heads will approach 0.5.
The proportion of heads will always be 1.
The proportion of heads will decrease to 0.
The proportion of heads will remain unpredictable.
What is the probability of rolling a 3 on a six-sided die?
1/6
1/3
1/2
1/12
What is the probability of rolling a 7 on a six-sided die?
0
1/6
1/7
1/12
What is the probability of rolling a number less than 5 on a six-sided die?
4/6
5/6
3/6
2/6
What is the sample space when rolling a six-sided die?
{1, 2, 3, 4, 5, 6}
{1, 2, 3, 4, 5, 6, 7}
{2, 4, 6, 8, 10, 12}
{1, 3, 5, 7, 9, 11}
How many possible outcomes are there when rolling two six-sided dice?
12
18
36
24
Which of the following is a possible outcome when rolling two dice?
1,7
2,5
7,1
0,6
If you roll two dice, what is the outcome when both dice show the same number?
1,2
2,3
3,3
4,5
What is the outcome when the first die shows 6 and the second die shows 4?
4,6
6,4
5,4
6,6
What is the probability that you would obtain a sum of 7 or a sum of 11 on the first roll of two dice?
0.222
0.111
0.500
0.333
What is the probability that you would obtain a sum of 2, 3, or 12 on the first roll of two dice?
0.111
0.222
0.250
0.333
What is the probability that you would roll again after your first roll, given the probabilities of getting a sum of 7 or 11, and a sum of 2, 3, or 12?
It is the complement of the probabilities of getting a sum of 7, 11, 2, 3, or 12.
It is the sum of the probabilities of getting a sum of 7, 11, 2, 3, or 12.
It is always 0.5.
It is always 1.
A company surveyed 1502 U.S. adults to determine how they read books during the past year. According to the results, how many adults read only print books?
560
409
425
108
Based on the survey of 1502 U.S. adults, what is the probability that the next adult surveyed read only print books during the last year?
560/1502
409/1502
425/1502
108/1502
In the survey, which category had the fewest number of adults?
Read only digital books
Read only print books
Read both print and digital books
Read no books
Which of the following is included in the category of digital books according to the survey?
Ebooks and audio books
Only ebooks
Only audio books
Printed magazines
A company surveyed 3000 users of a social media application to determine their ages. According to the frequency distribution, what is the probability that the next user surveyed is 25 to 34 years old?
0.255
0.153
0.765
0.459
Which type of probability is represented by the statement: "The probability that you will get an A on your next test is 0.9"?
Classical probability
Empirical probability
Subjective probability
Experimental probability
Classify the statement: "The probability that a voter chosen at random will be younger than 35 years old is 0.3." Is this an example of classical probability, empirical probability, or subjective probability?
Classical probability
Empirical probability
Subjective probability
Theoretical probability
Which type of probability is illustrated by the statement: "The probability of winning a 1000-ticket raffle with one ticket is 1/1000"?
Classical probability
Empirical probability
Subjective probability
Experimental probability
According to the range of probabilities rule, what is the possible range for the probability of an event E?
0 ≤ P(E) ≤ 1
-1 ≤ P(E) ≤ 1
0 ≤ P(E) ≤ 10
1 ≤ P(E) ≤ 2
What does a probability of 0 mean for an event?
The event is certain to happen
The event is impossible
The event is likely to happen
The event has an even chance of happening
On the probability scale, what value represents an "even chance" of an event occurring?
0
0.25
0.5
1
If an event is "certain" to happen, what is its probability?
0
0.25
0.75
1
What is the complement of event E in a sample space?
The set of all outcomes included in event E.
The set of all outcomes in a sample space that are not included in event E.
The set of all possible outcomes in event E and E'.
The set of all outcomes that are impossible.
How is the complement of event E usually denoted?
E''
E'
E*
E^
Which of the following equations correctly represents the relationship between the probability of event E and its complement?
P(E) + P(E') = 2
P(E) + P(E') = 0
P(E) + P(E') = 1
P(E) + P(E') = -1
If P(E) = 0.7, what is P(E')?
0.3
0.7
1.7
0.0
Which formula gives the probability of the complement of event E?
P(E') = 1 + P(E)
P(E') = P(E)
P(E') = 1 - P(E)
P(E') = P(E) - 1
What is the probability of randomly selecting a social networking site user who is NOT 25 to 34 years old, with a total of 3000 users?
0.745
0.255
0.500
0.765
If the total number of social networking site users is 3000, how many users are in the age group 25 to 34?
765
546
432
459
Which age group has the lowest frequency of social networking site users?
13 to 17
65 and over
55 to 64
45 to 54
What is the sum of the frequencies for users aged 35 to 44 and 45 to 54?
978
546
432
765
What is the probability of tossing a tail and spinning an odd number on the spinner shown?
1/4
1/8
1/2
1/16
If you toss a coin and spin the spinner, what is the probability of tossing a head or spinning a number greater than 3?
7/8
1/2
3/4
5/8
If your college identification number consists of eight digits, and each digit can be 0 through 9 (with repetition allowed), what is the probability of randomly generating your specific college identification number?
1081
8101
881
10101
What can the Multiplication Rule help you find?
The probability of two events occurring in sequence and conditional probabilities
The sum of two numbers
The speed of a moving object
The volume of a cylinder
What does conditional probability refer to?
The probability of an event occurring, given that another event has already occurred
The probability of two events occurring independently
The probability of an event not occurring
The probability of all possible events
How is the conditional probability of event B given event A usually denoted?
P(B | A)
P(A | B)
P(A ∩ B)
P(B) + P(A)
How should P(B | A) be read?
Probability of B, given A
Probability of A, given B
Probability of A and B
Probability of B or A
If two cards are selected in sequence from a standard deck and the first card is a king (not replaced), what is the probability that the second card is a queen?
4/51 ≈ 0.078
4/52 ≈ 0.077
1/13 ≈ 0.077
1/52 ≈ 0.019
Based on the table, what is the probability that an adult is 18 to 64 years old, given that the adult has ridden as a passenger in a self-driving vehicle?
202/225
202/751
202/970
202/219
How many adults aged 65 and older have ridden as a passenger in a self-driving vehicle according to the table?
23
202
196
225
What is the total number of U.S. adults surveyed in the poll?
970
751
225
219
According to the table, how many adults aged 18 to 64 have NOT ridden as a passenger in a self-driving vehicle?
549
202
196
745
What is the defining characteristic of independent events in probability?
The occurrence of one event does not affect the probability of the other event.
The occurrence of one event increases the probability of the other event.
The occurrence of one event decreases the probability of the other event.
The events must occur at the same time.
Which of the following equations correctly represents independent events?
P(B | A) = P(B)
P(B | A) = P(A)
P(A | B) = 0
P(A | B) = 1
If two events are not independent, they are called:
Dependent events
Mutually exclusive events
Random events
Impossible events
What is the multiplication rule for the probability of two events A and B occurring in sequence?
P(A and B) = P(A) × P(B | A)
P(A and B) = P(A) + P(B)
P(A and B) = P(A) × P(B)
P(A and B) = P(A) / P(B)
For independent events A and B, how can the multiplication rule for probability be simplified?
P(A and B) = P(A) × P(B)
P(A and B) = P(A) + P(B)
P(A and B) = P(A) / P(B)
P(A and B) = P(A) × P(B | A)
The multiplication rule for independent events can be extended for:
Any number of independent events
Only two independent events
Only dependent events
Only mutually exclusive events
If event A is driving over 85 miles per hour and event B is getting in a car accident, are these events independent or dependent?
Independent
Dependent
Mutually exclusive
Unrelated
When tossing a coin and getting a head (Event A), and then rolling a six-sided die and obtaining a 6 (Event B), are these two events independent or dependent?
Independent
Dependent
Mutually exclusive
Complementary
What is the probability of rolling a 6 on a six-sided die, regardless of the outcome of a coin toss?
1/6
1/2
1/12
1/3
If the occurrence of Event A does not change the probability of the occurrence of Event B, how are the events classified?
Independent
Dependent
Conditional
Exclusive
When selecting a king from a standard deck of 52 playing cards, not replacing it, and then selecting a queen from the deck, are the events independent or dependent?
Independent
Dependent
Mutually exclusive
Complementary
What is the probability of selecting a queen from a standard deck of 52 playing cards?
4/51
1/13
4/52
1/4
If the first card drawn from a deck is a king and is not replaced, what is the probability that the second card drawn is a queen?
4/52
1/13
4/51
1/4
When two cards are selected without replacement from a standard deck of 52 playing cards, what is the probability of selecting a king and then selecting a queen?
0.006
0.016
0.026
0.106
Why are the events of selecting a king and then a queen from a deck of cards without replacement considered dependent?
Because the first card is not replaced
Because the cards are shuffled after each draw
Because the deck contains jokers
Because the cards are replaced after each draw
What is the formula used to find the probability of selecting a king and then a queen without replacement from a deck of cards?
P(K and Q) = P(K) × P(Q|K)
P(K and Q) = P(K) + P(Q)
P(K and Q) = P(K) × P(Q)
P(K and Q) = P(K|Q) × P(Q)
What is the value of P(K) when selecting a king from a standard deck of 52 playing cards?
4/52
1/13
1/4
13/52
After selecting a king and not replacing it, what is the probability of selecting a queen from the remaining cards?
4/51
4/52
1/13
1/51
If a coin is tossed and a die is rolled, what is the probability of getting a head on the coin and then rolling a 6 on the die?
1/12
1/6
1/2
1/36
When tossing a coin and rolling a die, are the events of getting a head and rolling a 6 independent?
Yes, they are independent.
No, they are dependent.
They are mutually exclusive.
They are complementary.
What is the decimal approximation of the probability of tossing a head and then rolling a 6?
0.083
0.5
0.167
0.25
What is the probability that three anterior cruciate ligament (ACL) surgeries are all successful, given that the probability of success for each surgery is 0.95?
0.857
0.950
0.900
0.995
If the probability of a successful ACL surgery is 0.95, what is the probability that all three surgeries are successful, assuming each surgery is independent?
(0.95)(0.95)(0.95)
(0.95)+(0.95)+(0.95)
(0.95)/(0.95)/(0.95)
(0.95)-(0.95)-(0.95)
According to the example, the probability of success for one ACL surgery is:
0.95
0.85
0.75
0.65
What is the probability that an anterior cruciate ligament (ACL) reconstructive surgery is successful?
0.95
0.05
0.50
0.90
If the probability of success for one ACL surgery is 0.95, what is the probability of failure for one surgery?
0.05
0.95
0.50
0.10
What is the probability that none of the three ACL surgeries are successful?
0.0001
0.05
0.95
0.001
What is the probability that a randomly selected senior who was matched to a residency position did NOT get matched with one of the senior’s top three choices?
0.244
0.756
0.500
0.100
Which rule is used to find the probability that a senior did not get matched with one of their top three choices?
The complement rule
The addition rule
The subtraction rule
The division rule
If $ P(B | A) = 0.756 $, what is $ P(B' | A) $?
0.244
0.756
1.000
0.500
What is the probability that a randomly selected senior was matched with a residency position?
18,108/19,326
0.756
0.708
0.500
Given that a senior was matched to a residency position, what is the probability that it was one of the senior’s top three choices?
0.756
0.500
0.708
18,108/19,326
What is the probability that a randomly selected senior was matched with a residency position and it was one of the senior’s top three choices?
0.708
0.756
0.500
18,108/19,326
Which rule is used in the example to find the probability that two events both occur?
Multiplication Rule
Addition Rule
Subtraction Rule
Division Rule
In a recent year, how many U.S. MD medical school seniors applied to residency programs?
19,326
18,108
75,600
10,000
What percentage of U.S. MD medical school seniors were matched with one of their top three residency program choices?
75.6%
50%
90%
60%
What does the term "match" refer to in the context of medical residency programs?
The process where a student's and a program director's preference lists overlap, resulting in placement
A test taken by medical students
A scholarship program for medical students
A type of medical specialty
Who ranks the residency programs in order of preference?
Medical students
Program directors
Hospital administrators
Nurses
Who ranks the students applying to residency programs in the United States?
Program directors
Medical students
Patients
Government officials
What is the probability that an anterior cruciate ligament (ACL) reconstructive surgery is successful?
0.95
0.50
0.75
0.99
If three ACL surgeries are performed, what is the probability that at least one of them is successful?
0.9999
0.9500
0.9000
0.5000
Which rule is used to find the probability that at least one of the three ACL surgeries is successful?
Complement rule
Addition rule
Subtraction rule
Division rule
What does the event “at least one successful” mean in probability terms?
One or more are successful
None are successful
All are unsuccessful
Only one is unsuccessful
According to the example, would it be unusual for a randomly selected senior to be matched with a residency position that was one of the senior’s top three choices?
No, because the probability is about 0.708, which is greater than 0.05.
Yes, because the probability is less than 0.05.
Yes, because the probability is exactly 0.05.
No, because the probability is less than 0.05.
What is the probability that a senior is matched with a residency position that was one of their top three choices?
0.708
0.05
0.5
0.08
If the probability of an event is greater than 0.05, how is the event described in the robust example?
Likely to happen
Unlikely to happen
Impossible to happen
Certain to happen
Which of the following is an objective of Section 3.3?
How to determine whether two events are mutually exclusive
How to solve quadratic equations
How to calculate the mean of a dataset
How to graph linear functions
What rule is used in Section 3.3 to find the probability of two events?
Addition Rule
Multiplication Rule
Subtraction Rule
Division Rule
What does it mean for two events to be mutually exclusive?
They cannot occur at the same time
They always occur together
They have some outcomes in common
They are independent
If events A and B have no outcomes in common, what can be said about them?
They are mutually exclusive
They are dependent
They are independent
They always occur together
Which of the following statements is true about mutually exclusive events?
They cannot occur at the same time
They always have at least one outcome in common
They are always independent
They must occur together
If Event A is rolling a 3 on a die and Event B is rolling a 4 on a die, are these events mutually exclusive?
Yes, they are mutually exclusive.
No, they are not mutually exclusive.
Yes, they can occur at the same time.
No, they are independent events.
Why are the events "rolling a 3 on a die" and "rolling a 4 on a die" considered mutually exclusive?
Because both outcomes cannot occur at the same time.
Because both outcomes always occur together.
Because they are dependent events.
Because they have the same outcome.
What is the probability of selecting a card that is either a 4 or an ace from a standard deck of 52 cards?
0.154
0.250
0.077
0.308
If you select a card from a standard deck, how many cards are either a 4 or an ace?
8
4
12
2
When finding the probability of drawing a 4 or an ace from a deck, why can you simply add the probabilities?
Because the events are mutually exclusive
Because the events are independent
Because the deck has only 4s and aces
Because the probabilities are always equal
What is the probability of drawing a 4 from a standard deck of 52 cards?
4/52
8/52
1/52
0.154
What is the addition rule for the probability of events A or B occurring?
P(A or B) = P(A) + P(B) – P(A and B)
P(A or B) = P(A) × P(B)
P(A or B) = P(A) – P(B)
P(A or B) = P(A) ÷ P(B)
For mutually exclusive events A and B, how is the addition rule for probability simplified?
P(A or B) = P(A) + P(B)
P(A or B) = P(A) × P(B)
P(A or B) = P(A) – P(B)
P(A or B) = P(A) ÷ P(B)
Which of the following statements is true about mutually exclusive events?
The addition rule can be extended to any number of mutually exclusive events.
Mutually exclusive events always occur together.
The probability of mutually exclusive events is always zero.
Mutually exclusive events cannot be added.
Are the following events mutually exclusive? Event A: Randomly select a blood donor with type O blood. Event B: Randomly select a female blood donor.
Yes, they are mutually exclusive.
No, they are not mutually exclusive.
They are always independent.
They are complementary events.
If Event A is "Randomly select a male student" and Event B is "Randomly select a nursing major," are these events mutually exclusive? Explain your reasoning.
Yes, because a male student cannot be a nursing major.
No, because a student can be a male nursing major.
Yes, because nursing majors are only female.
No, because all students are nursing majors.
When rolling a die, what is the probability of rolling a number less than 3 or rolling an odd number?
4/6
5/6
1/2
1/3
According to the example, are the events "rolling a number less than 3" and "rolling an odd number" mutually exclusive when rolling a die?
Yes, they are mutually exclusive.
No, they are not mutually exclusive.
They are independent events.
They are complementary events.
Which number is an outcome of both the events "rolling a number less than 3" and "rolling an odd number" when rolling a die?
2
3
1
5
What is the total number of months recorded in the frequency distribution table for sales volumes?
36
30
24
40
According to the frequency distribution, how many months did the sales representative achieve sales between $75,000 and $99,999?
7
6
9
5
How many months did the sales representative achieve sales between $100,000 and $124,999?
9
7
6
5
What is the probability that the sales representative will sell between $75,000 and $124,999 next month, based on the given data?
16/36
7/36
9/36
12/36
Which sales volume range had the least number of months?
$175,000–199,999
$0–24,999
$125,000–149,999
$150,000–174,999
What is the probability that a randomly selected donor has type O or type A blood?
0.85
0.80
0.85
0.70
According to the table, how many donors have type B blood?
45
37
8
409
How many donors are Rh-negative according to the table?
65
45
409
344
What is the total number of donors represented in the table?
409
344
65
45
According to the solution, why are the events "type B blood" and "Rh-negative" not mutually exclusive?
Because a donor can have type B blood and be Rh-negative
Because all donors are either type B or Rh-negative
Because type B blood is always Rh-positive
Because Rh-negative donors cannot have type B blood
What is a permutation?
An ordered arrangement of objects
An unordered arrangement of objects
A type of equation
A random selection of objects
What is the formula for the number of permutations of n distinct objects?
n!
n2
n + 1
2n
What is the value of 0! (zero factorial)?
1
0
Undefined
n
What is the value of 6! (six factorial)?
720
120
36
24
What is the value of 4! (four factorial)?
24
12
16
20
Based on the figure, what is the probability that a randomly selected draft pick from the 2020 NFL draft is NOT a running back or a wide receiver?
202/255
53/255
16/255
37/255
What is the formula for classical probability?
P(E) = Number of outcomes in event E / Number of outcomes in sample space
P(E) = Frequency of event E / Total frequency
P(E) = 1 - P(E')
P(A and B) = P(A) * P(B|A)
Which type of probability is estimated from experimentation?
Empirical Probability
Classical Probability
Complementary Events
Addition Rule
What is the range of probabilities for any event?
0 ≤ P(E) ≤ 1
0 ≤ P(E) ≤ 10
-1 ≤ P(E) ≤ 1
0 ≤ P(E) ≤ 100
What is the formula for the probability of the complement of event E?
P(E') = 1 - P(E)
P(E') = P(E) + 1
P(E') = P(E) * 2
P(E') = 1 / P(E)
Which formula is used to find the probability of two independent events both occurring?
P(A and B) = P(A) * P(B)
P(A and B) = P(A) + P(B)
P(A and B) = P(A) * P(B|A)
P(A and B) = P(A) - P(B)
What is the addition rule for the probability of at least one of two events occurring?
P(A or B) = P(A) + P(B) - P(A and B)
P(A or B) = P(A) * P(B)
P(A or B) = P(A) - P(B)
P(A or B) = P(A) / P(B)
For mutually exclusive events, what is the formula for the probability of A or B?
P(A or B) = P(A) + P(B)
P(A or B) = P(A) * P(B)
P(A or B) = P(A) - P(B)
P(A or B) = P(A) / P(B)
What is the objective of a 9 x 9 Sudoku number puzzle?
To fill the grid so that each row, column, and 3 x 3 grid contain the digits 1 to 9
To fill the grid with only even numbers
To fill the grid with only odd numbers
To fill the grid so that each row contains the same digit
How many different ways can the first row of a blank 9 x 9 Sudoku grid be filled?
362,880 ways
81 ways
9,999 ways
1,234 ways
What is the value of 9 factorial (9!)?
362,880
9,000
81
45,360
If you have 11 letters consisting of one M, four I’s, four S’s, and two P’s, what is the probability that a random arrangement of these letters spells the word "Mississippi"?
1 divided by (11! / (1! 4! 4! 2!))
1 divided by 11!
1 divided by (4! 4! 2!)
1 divided by (1! 2! 3! 5!)
A building contractor is planning to develop a subdivision that consists of 6 one-story houses, 4 two-story houses, and 2 split-level houses. In how many distinguishable ways can the houses be arranged?
13,860 distinguishable ways
12,000 distinguishable ways
8,640 distinguishable ways
24,000 distinguishable ways
Given 12 houses in a subdivision, with 6 of one type, 4 of another, and 2 of a third type, which formula would you use to find the number of distinguishable arrangements?
12! / (6! 4! 2!)
12! / (3! 3! 3! 3!)
12! / (4! 4! 4!)
12! / (2! 2! 2! 2! 2! 2!)
If a subdivision has 12 houses, and the numbers of each type are n₁ = 6, n₂ = 4, n₃ = 2, what is the value of n?
12
6
4
2
What does a combination of n objects taken r at a time represent?
A selection of r objects from a group of n objects without regard to order
A selection of n objects from a group of r objects with regard to order
A selection of r objects from a group of n objects with regard to order
A selection of n objects from a group of r objects without regard to order
Which of the following is the correct formula for the number of combinations of n objects taken r at a time?
nCr = n! / [(n - r)! r!]
nCr = n! / (n - r)!
nCr = n! / r!
nCr = (n - r)! / n! r!
In a state's department of transportation project, 16 companies bid for a new section of interstate highway. If the state plans to hire four companies, how many companies are being selected from the group?
2
3
4
5
When selecting four companies from a group of 16 for a project, which of the following statements is true?
Order is important
Order is not important
Only one company can be selected
All companies must be selected
If you need to select 4 companies from a group of 16, what are the values of n and r in the combination formula?
n = 4, r = 16
n = 16, r = 4
n = 8, r = 2
n = 20, r = 5
A student advisory board consists of 17 members. Three members serve as the board’s chair, secretary, and webmaster. Each member is equally likely to serve any of the positions. What is the probability of selecting at random the three members that hold each position?
1 / 4080
1 / 4913
1 / 136
1 / 17
What is the formula for the number of distinguishable permutations of n objects where n₁ are of one type, n₂ are of another type, and so on?
n! / (n₁! n₂! n₃! ... nₖ!)
n! * (n₁! n₂! n₃! ... nₖ!)
n! + (n₁! n₂! n₃! ... nₖ!)
n! - (n₁! n₂! n₃! ... nₖ!)
In the formula for distinguishable permutations, what must be true about the sum of n₁, n₂, n₃, ..., nₖ?
n₁ + n₂ + n₃ + ... + nₖ = n
n₁ + n₂ + n₃ + ... + nₖ = 0
n₁ + n₂ + n₃ + ... + nₖ = 1
n₁ + n₂ + n₃ + ... + nₖ = n₁
Which of the following best describes a distinguishable permutation?
A permutation where objects of the same type are considered identical
A permutation where all objects are unique
A permutation with only one type of object
A permutation with no repeated objects
How many ways are there to form a four-digit code using digits 0-9 if no digit is repeated?
5040
10000
4096
720
If you need to select 4 digits from a group of 10 without repetition, which formula should you use?
nCr
nPr
n!
r!
What is the value of 10P4?
720
4096
5040
10000
In the formula for permutations, what does "n" represent?
The number of items to select
The total number of items in the group
The number of repeated items
The number of arrangements
What is the formula for the number of permutations of n distinct objects taken r at a time?
n! / (n-r)!
n! / r!
nr
r! / (n-r)!
In the formula for permutations, what does the notation nPr represent?
The number of permutations of n objects taken r at a time
The number of combinations of n objects taken r at a time
The probability of selecting r objects from n
The sum of n objects taken r at a time
According to the formula for permutations, what is the condition for r and n?
r ≤ n
r > n
r=n2
r ≥ n
What is the objective of a 9 x 9 Sudoku number puzzle?
To fill the grid so that each row, each column, and each 3 x 3 grid contain the digits 1 to 9.
To fill the grid with only even numbers.
To fill the grid so that each row contains only one digit.
To fill the grid randomly with any numbers.
How many different ways can the first row of a blank 9 x 9 Sudoku grid be filled?
362,880 ways
81 ways
9 ways
40,320 ways
What is the value of 9 factorial (9!)?
362,880
40,320
720
3,628
