1 Million+ Step-by-step solutions

Each of the digits 0–9 is written on a slip of paper, and the slips are placed in a hat. If three slips of paper are selected at random, determine the probability that the three numbers selected are greater than 5.

The problems are to be done without re-placement. Use combinations to determine probabilities.

Barnes & Noble has 20 different books listed on its clearance list. Twelve books are listed as mystery, and 8 are listed as fantasy. If 4 books are selected at random from the list, determine the probability that they are all fantasy books.

The problems are to be done without re-placement. Use combinations to determine probabilities.

The sales department at Atwell Studios consists of three people, the manufacturing department consists of six people, and the accounting department consists of two people. Three people will be selected at random from these people and will be given gift certificates to Sweet Tomatoes, a local restaurant. Determine the prob-ability that two of those selected will be from the manufacturing department and one will be from the accounting department.

The problems are to be done without re-placement. Use combinations to determine probabilities.

A bicycle club has 10 members. Six members ride Diamondback bicycles, 2 members ride Fuji bicycles, and 2 members ride Schwinn bicycles. If 4 of the members are selected at random, determine the probability that all 4 ride Diamondback bicycles.

The problems are to be done without re-placement. Use combinations to determine probabilities.

You are dealt 5 cards from a standard deck of 52 cards. Determine the probability you are dealt 3 kings and 2 queens.

The problems are to be done without re-placement. Use combinations to determine probabilities.

You are dealt 5 cards from a standard deck of 52 cards. Determine the probability that you are dealt 5 red cards.

The problems are to be done without re-placement. Use combinations to determine probabilities.

A television game show has five doors, of which the contestant must pick two. Behind two of the doors are expensive cars, and behind the other three doors are consolation prizes. The contestant gets to keep the items behind the two doors she selects. Determine the probability that the contestant wins

No cars.

A television game show has five doors, of which the contestant must pick two. Behind two of the doors are expensive cars, and behind the other three doors are consolation prizes. The contestant gets to keep the items behind the two doors she selects. Determine the probability that the contestant wins

Exactly one car.

Assume that a particular professional baseball team has 10 pitchers, 6 infielders, and 9 other players. If 3 players’ names are selected at random from the team’s roster, determine the probability that.

All 3 are infielders.

Assume that a particular professional baseball team has 10 pitchers, 6 infielders, and 9 other players. If 3 players’ names are selected at random from the team’s roster, determine the probability that.

None of the three is a pitcher.

Assume that a particular professional baseball team has 10 pitchers, 6 infielders, and 9 other players. If 3 players’ names are selected at random from the team’s roster, determine the probability that.

2 are pitchers and 1 is an infielder.

1 is a pitcher and 2 are players other than pitchers and infielders.

A car rental agency has 10 midsized and 15 compact cars on its lot, from which 6 will be selected. Assuming that each car is equally likely to be selected and the cars are selected at random, determine the probability (as a decimal number rounded to four decimal places) that the cars selected consist of.

All midsized cars.

A car rental agency has 10 midsized and 15 compact cars on its lot, from which 6 will be selected. Assuming that each car is equally likely to be selected and the cars are selected at random, determine the probability (as a decimal number rounded to four decimal places) that the cars selected consist of.

2 midsized cars and 4 compact cars.

A car rental agency has 10 midsized and 15 compact cars on its lot, from which 6 will be selected. Assuming that each car is equally likely to be selected and the cars are selected at random, determine the probability (as a decimal number rounded to four decimal places) that the cars selected consist of.

3 midsized cars and 3 compact cars.

General Mills is testing 12 new cereals for possible production. They are testing 3 oat cereals, 4 wheat cereals, and 5 rice cereals. If each of the 12 cereals has the same chance of being produced, and 4 new cereals will be produced, determine the probability that the 4 new cereals that will be produced are as follows.

1 is a wheat cereal and 3 are rice cereals.

General Mills is testing 12 new cereals for possible production. They are testing 3 oat cereals, 4 wheat cereals, and 5 rice cereals. If each of the 12 cereals has the same chance of being produced, and 4 new cereals will be produced, determine the probability that the 4 new cereals that will be produced are as follows.

2 are oat cereals, 1 is a wheat cereal, and 1 is a rice cereal.

General Mills is testing 12 new cereals for possible production. They are testing 3 oat cereals, 4 wheat cereals, and 5 rice cereals. If each of the 12 cereals has the same chance of being produced, and 4 new cereals will be produced, determine the probability that the 4 new cereals that will be produced are as follows.

At least one is an oat cereal.

Of 26 employees at Wegmans supermarket, 15 work as cashiers and 11 stock shelves. If 3 of the 26 employees are selected at random to work overtime, determine the probability that all 3 are cashiers.

A full house in poker consists of three of one kind and two of another kind in a five-card hand. For example, if a hand contains three kings and two 5’s, it is a full house. If 5 cards are dealt at random from a standard deck of 52 cards, without replacement, determine the probability of getting three kings and two 5’s.

On a die, the sum of the dots on the opposite faces is seven. Two six-sided dice are placed together on top of one another, on a table, as shown in the figure below. The top and bottom faces of the bottom die and the bottom face of the top die cannot be seen. If you walk around the table, what is the sum of all the dots on all the visible faces of the dice?

Consider the following graph, which shows the U.S. population in 2000 and the projected U.S. population in 2050.

(a) Compute the projected percent increase in population from 2000 to 2050 by using the formula .

(b) Measure the radius and then compute the area of the circle representing 2000. Use A = πr^{2}.

(c) Repeat part (b) for the circle representing 2050.

(d) Compute the percent increase in the size of the area of the circle from 2000 to 2050.

(e) Are the circle graphs misleading?

Fill in the blanks with an appropriate word, phrase, or symbol(s).

A path that passes through each edge of a graph exactly one time is called a(n) ___________ path.

Fill in the blank with an appropriate word, phrase, or symbol(s).

When group A loses an item or items to group B even though group A’s population grew at a faster rate than group B’s, the _________ paradox occurs.

Fill in the blank with an appropriate word, phrase, or symbol(s).

When the addition of a new group and additional items to be apportioned reduces the prior apportionment of another group, the _________ paradox occurs.

Fill in the blank with an appropriate word, phrase, or symbol(s).

When an increase in the total number of items to be apportioned results in a loss of an item for a group, the _________ paradox occurs.

Fill in the blank with an appropriate word, phrase, or symbol(s).

Hamilton’s and Jefferson’s apportionment methods, favor ____________ states.

Fill in the blank with an appropriate word, phrase, or symbol(s).

Adams’and Webster’s apportionment methods favor ___________ states.

Fill in the blank with an appropriate word, phrase, or symbol(s).

The apportionment method that satisfies the quota rule but may produce a paradox is called __________ method.

Consider the apportionment of 60 doctors for First Physicians Organization. The apportionment using Hamilton’s method is shown in the table below.

Does the Alabama paradox occur using Hamilton’s method if the number of doctors is increased from 60 to 61? Explain your answer.

When appropriate, round quotas and divisors to the nearest hundredth.

A large company with offices in four cities must distribute 144 new ergonomic chairs to the four offices. The chairs will be apportioned based on the number of employees in each office as shown in the table below.

(a) Apportion the chairs using Hamilton’s method.

(b) Does the Alabama paradox occur using Hamilton’s method if the number of new chairs is increased from 144 to 145? Explain your answer.

When appropriate, round quotas and divisors to the nearest hundredth.

A country with three states has 30 seats in the legislature. The population of each state is shown in the table below.

(a) Apportion the seats using Hamilton’s method.

(b) Does the Alabama paradox occur using Hamilton’s method if the number of seats is increased from 30 to 31? Explain your answer.

When appropriate, round quotas and divisors to the nearest hundredth.

A television game show has five doors, of which the contestant must pick two. Behind two of the doors are expensive cars, and behind the other three doors are consolation prizes. The contestant gets to keep the items behind the two doors she selects. Determine the probability that the contestant wins

Both cars.

A country with three states has 200 seats in the legislature. The population of each state is shown in the table below.

(a) Apportion the seats using Hamilton’s method.

(b) Does the Alabama paradox occur using Hamilton’s method if the number of seats is increased from 200 to 201? Explain your answer.

When appropriate, round quotas and divisors to the nearest hundredth.

AT&T has 25,000 employees in three cities as shown in the table below. It wishes to give promotions to 200 employees.

(a) Apportion the promotions using Hamilton’s method.

(b) Suppose that in 10 years the cities have the following number of employees and the company wishes to again give promotions to 200 employees. Does the population paradox occur using Hamilton’s method?

When appropriate, round quotas and divisors to the nearest hundredth.

Anabru Manufacturing has 100 forklift trucks to apportion among 3 factories. The trucks are to be apportioned based on the number of employees at each factory as shown in the table below

(a) Apportion the trucks using Hamilton’s method.

(b) Suppose that 1 year later the factories have the following number of employees. If the 100 forklift trucks are reap-portioned to the factories, does the population paradox occur using Hamilton’s method?

When appropriate, round quotas and divisors to the nearest hundredth.

A college with five divisions has funds for 54 internships. The student population for each division is shown in the table below.

(a) Apportion the internships using Hamilton’s method.

(b) Suppose that 1 year later the divisions have the following populations. If the college wishes to apportion 54 internships, does the population paradox occur using Hamilton’s method?

When appropriate, round quotas and divisors to the nearest hundredth.

Suffolk County Community College is holding an election to appoint a chairperson of the board of trustees. The 413 faculty members vote as follows: Michelle, 231 votes; Jeffrey, 155 votes; and Donald, 27 votes.

(a) Using the plurality method, who is elected?

(b) Did this candidate receive a majority of votes?

Fill in the blank with an appropriate word, phrase, or symbol(s).

A candidate who wins a first election, then gains additional support without losing any of the original support, should also win a second election. This criterion is called the _________ criterion.

Fill in the blank with an appropriate word, phrase, or symbol(s).

A standard quota rounded up to the nearest integer is called a(n) ________ quota.

Ten voters are asked to rank four candidates. The 10 voters turn in the following ballots showing their preferences in order:

Make a preference table for these ballots.

Fill in the blank with an appropriate word, phrase, or symbol(s).

If a candidate is favored when compared head-to-head with every other candidate in an election, that candidate should be declared the winner. This criterion is called the _________ criterion.

Fill in the blank with an appropriate word, phrase, or symbol(s).

A standard quota rounded down to the nearest integer is called a(n) ____________ quota.

Seven voters are asked to rank three candidates. The seven voters turn in the following ballots showing their preferences in order:

Make a preference table for these ballots.

Fill in the blank with an appropriate word, phrase, or symbol(s).

If a candidate is the winner of an election and in a second election one or more of the other candidates is removed, the previous winner should still be the winner. This criterion is called the ___________ criterion.

Fill in the blank with an appropriate word, phrase, or symbol(s).

The rule stating that an apportionment should always be either the upper quota or the lower quota is called the _________ rule.

How many members voted?

The members of the Student Council at Ohio State University are planning to go out to dinner following an upcoming meeting. The restaurant choices are Chipotle Mexican Grill (C), Jimmy John’s (J), Domino’s Pizza (D), and Burger King (B). The members rank their choices according to the following preference table.

Fill in the blank with an appropriate word, phrase, or symbol(s).

A voting method that may not satisfy any of the fairness criteria is the ____________ method.

Fill in the blank with an appropriate word, phrase, or symbol(s).

Jefferson’s method, Webster’s method, and Adams’ method require using a(n) __________ quota.

Using the plurality method, which restaurant is chosen?

The members of the Student Council at Ohio State University are planning to go out to dinner following an upcoming meeting. The restaurant choices are Chipotle Mexican Grill (C), Jimmy John’s (J), Domino’s Pizza (D), and Burger King (B). The members rank their choices according to the following preference table.

Fill in the blank with an appropriate word, phrase, or symbol(s).

A voting method that always satisfies the majority criterion and head-to-head criterion, but may not satisfy any other criterion, is the _________ method.

Fill in the blank with an appropriate word, phrase, or symbol(s).

The apportionment method that requires rounding the standard quota down to the lower quota is called ________ method.

Using the Borda count method, which restaurant is chosen?

The members of the Student Council at Ohio State University are planning to go out to dinner following an upcoming meeting. The restaurant choices are Chipotle Mexican Grill (C), Jimmy John’s (J), Domino’s Pizza (D), and Burger King (B). The members rank their choices according to the following preference table.

Fill in the blank with an appropriate word, phrase, or symbol(s).

A voting method that always satisfies the majority criterion but may not satisfy any other criterion is the ____________ method.

Fill in the blank with an appropriate word, phrase, or symbol(s).

(a) The apportionment method that uses a modified divisor that is less than the standard divisor is ________ method.

(b) The apportionment method that uses a modified divisor that is greater than the standard divisor is _________ method.

(c) The apportionment method that uses a modified divisor that could be less than, greater than, or equal to the standard divisor is _________ method.

Using the plurality with elimination method, which restaurant is chosen?

Fill in the blank with an appropriate word, phrase, or symbol(s).

A voting method that always satisfies the majority criterion and the monotonicity criterion, but may not satisfy any other criterion, is the _________ method.

Fill in the blank with an appropriate word, phrase, or symbol(s).

(a) The apportionment method that uses a modified quota that is always rounded to the nearest integer is ________ method.

(b) The apportionment method that uses a modified quota that is always rounded up to the nearest integer is _________ method.

(c) The apportionment method that uses a modified quota that is always rounded down to the nearest integer is ________ method.

Using the pairwise comparison method, which restaurant is chosen?

Members of the board of directors of the American Nursing Association are voting to select a city to host the association’s annual meeting. The board is considering the following cities: Portland (P), Orlando (O), and Nashville (N). The preference table for the 15 members of the board of directors is shown below. Show that the Borda count method violates the majority criterion.

Fill in the blank with an appropriate word, phrase, or symbol(s).

Jefferson’s method, Webster’s method, and Adams’ method all make use of a modified quota and can all lead to violations of the __________________ rule.

Which restaurant is chosen if the plurality with elimination method is used and the restaurant with the most last-place votes is eliminated at each step?

(a) Determine the standard divisor.

(b) Determine each state’s standard quota.

When appropriate round quotas to the nearest hundredth.

Suppose that Turtlestan is a small country with a population of 8,000,000 that consists of four states, A, B, C, and D. There are 160 seats in the legislature that need to be apportioned among the four states. The population of each state is shown in the table below.

How many employees voted?

The employees at Delphi Engineering must decide whether to play baseball (B), soccer (S), or volleyball (V) at their year-end picnic. The preference table follows.

Determine each state’s apportionment using Hamilton’s method.

When appropriate round quotas to the nearest hundredth.

Suppose that Turtlestan is a small country with a population of 8,000,000 that consists of four states, A, B, C, and D. There are 160 seats in the legislature that need to be apportioned among the four states. The population of each state is shown in the table below.

Determine the winner using the plurality method.

The employees at Delphi Engineering must decide whether to play baseball (B), soccer (S), or volleyball (V) at their year-end picnic. The preference table follows.

(a) Determine each state’s modified quota using the divisor 49,300.

(b) Determine each state’s apportionment using Jefferson’s method.

When appropriate round quotas to the nearest hundredth.

Suppose that Turtlestan is a small country with a population of 8,000,000 that consists of four states, A, B, C, and D. There are 160 seats in the legislature that need to be apportioned among the four states. The population of each state is shown in the table below.

Determine the winner using the Borda count method.

The employees at Delphi Engineering must decide whether to play baseball (B), soccer (S), or volleyball (V) at their year-end picnic. The preference table follows.

A country with three states has 250 seats in the legislature. The population of each state is shown in the table below.

(a) Apportion the seats using Hamilton’s method.

(b) Suppose that in 10 years the states have the following populations. If the country reapportions 250 seats in the legislature, does the population paradox occur using Hamilton’s method?

When appropriate, round quotas and divisors to the nearest hundredth.

(a) Determine each state’s modified quota using the divisor 49,250.

(b) Determine each state’s apportionment using Jefferson’s method.

When appropriate round quotas to the nearest hundredth.

Determine the winner using the plurality with elimination method.

Cynergy Telecommunications has employees in Europe, labeled A, and in the United States, labeled B. The number of employees in each group is shown in the table below. There are 48 managers to be apportioned between the two groups.

(a) Apportion the managers using Hamilton’s method.

(b) Suppose that additional employees in Asia, labeled C, with the number of employees shown in the table below, are added with seven new managers. Does the new-states paradox occur using Hamilton’s method?

When appropriate, round quotas and divisors to the nearest hundredth.

(a) Determine each state’s modified quota using the divisor 50,600.

(b) Determine each state’s apportionment using Adams’ method.

When appropriate round quotas to the nearest hundredth.

Determine the winner using the pairwise comparison method.

The town of Manlius purchased 25 new picnic tables to be apportioned between two parks. The picnic tables are to be apportioned based on the annual number of visitors to each park as shown below.

(a) Apportion the picnic tables using Hamilton’s method.

(b) Suppose that the town decides to purchase five additional picnic tables and include a third park with an annual number of visitors as shown in the table below. The town will now apportion 30 picnic tables among the three parks. Does the new-states paradox occur using Hamilton’s method?

When appropriate, round quotas and divisors to the nearest hundredth.

(a) Determine each state’s modified quota using the divisor 50,600.

(b) Determine each state’s apportionment using Adams’ method.

When appropriate round quotas to the nearest hundredth.

Determine the winner if the plurality with elimination method is used and the candidate with the most last-place votes is eliminated at each step.

A country with two states has 33 seats in the legislature. The population of each state is shown in the table below.

(a) Apportion the states using Hamilton’s method.

(b) Suppose that a third state with the population shown in the table below is added, with seven additional seats. Does the new-states paradox occur using Hamilton’s method?

When appropriate, round quotas and divisors to the nearest hundredth.

Determine each state’s apportionment using Webster’s method using the standard divisor.

When appropriate round quotas to the nearest hundredth. Suppose that Turtlestan is a small country with a population of 8,000,000 that consists of four states, A, B, C, and D. There are 160 seats in the legislature that need to be apportioned among the four states. The population of each state is shown in the table below.

A country with three states has 66 seats in the legislature. The population of each state is shown in the table at the top of the right column.

(a) Apportion the seats using Hamilton’s method.

(b) Suppose that a fourth state with the population shown in the table below is added, with five additional seats. Does the new-states paradox occur using Hamilton’s method?

When appropriate, round quotas and divisors to the nearest hundredth.

(a) Determine each state’s modified quota using the divisor 49,900.

(b) Determine each state’s apportionment using Webster’s method.

When appropriate round quotas to the nearest hundredth. Suppose that Turtlestan is a small country with a population of 8,000,000 that consists of four states, A, B, C, and D. There are 160 seats in the legislature that need to be apportioned among the four states. The population of each state is shown in the table below.

(a) Determine the standard divisor.

(b) Determine each hotel’s standard quota.

When appropriate round quotas to the nearest hundredth.

A large hotel chain needs to apportion 25 new staff members among three hotels based on the numbers of rooms in each hotel as shown in the table below.

Determine each resort’s apportionment using Jefferson’s method. (Some divisors between 11 and 12 will work.)

When appropriate round quotas to the nearest hundredth.

Sandy Shores Resorts operates four beach resorts in the Caribbean islands. Sandy Shores Resorts plans to apportion 50 new beach umbrellas among the four resorts based on the number of rooms in each resort as shown in the table below.

Consider the preference table below. Assume that a majority is needed to win the election. Since no candidate has a majority but C has the most first-place votes, a runoff election is held between A and B. The winner of the runoff election will run against C.

(a) Using the plurality method, which candidate will win the runoff election, A or B?

(b) Will the candidate determined in part (a) win the election against candidate C?

(c) Suppose that the voters who support candidate C, the candidate with the most first-place votes, find out prior to the vote that C will not win if B is eliminated and C runs against A. How can the voters who support C vote insincerely to enable C to win?

Determine each route’s apportionment using Adams’ method.

When appropriate round quotas to the nearest hundredth.

The Transit Department in the city of Houston has 100 new buses to be apportioned among six routes. The department decides to apportion the buses based on the average number of daily passengers per route, as shown in the table below.

Determine each route’s apportionment using Webster’s method.

When appropriate round quotas to the nearest hundredth.

The Transit Department in the city of Houston has 100 new buses to be apportioned among six routes. The department decides to apportion the buses based on the average number of daily passengers per route, as shown in the table below.

(a) Determine the standard divisor.

(b) Determine each shift’s standard quota.

(c) Determine each shift’s apportionment using Hamilton’s method.

When appropriate round quotas to the nearest hundredth.

A hospital has 200 nurses to be apportioned among four shifts: shifts A, B, C, and D. The hospital decides to apportion the nurses based on the average number of room calls reported during each shift. Room calls are shown in the table at the top of the right column.

Determine each shift’s apportionment using Adams’ method. (Divisors do not have to be whole numbers.)

When appropriate round quotas to the nearest hundredth.

A hospital has 200 nurses to be apportioned among four shifts: shifts A, B, C, and D. The hospital decides to apportion the nurses based on the average number of room calls reported during each shift. Room calls are shown in the table at the top of the right column.

Determine each shift’s apportionment using Jefferson’s method. (Divisors do not have to be whole numbers.)

When appropriate round quotas to the nearest hundredth.

A hospital has 200 nurses to be apportioned among four shifts: shifts A, B, C, and D. The hospital decides to apportion the nurses based on the average number of room calls reported during each shift. Room calls are shown in the table at the top of the right column.

Determine each shift’s apportionment using Webster’s method. (Some divisors between 12 and 12.05 will work.)

When appropriate round quotas to the nearest hundredth.

In 1970, the first United States census was taken. The following table shows the population of the 15 states at that time. One hundred five seats in the United States House of Representatives were to be apportioned among the 15 states.

(a) Determine the apportionment that would have resulted if Hamilton’s method had been used as the method originally approved by Congress.

(b) Determine the apportionment that was used with Jefferson’s method.

(c) Compare the apportionments from parts (a) and (b). Which state(s) benefited from Jefferson’s method? Which state(s) were at a disadvantage from Jefferson’s method?

When appropriate round quotas to the nearest hundredth.

Suppose that a country with a population of 10,000,000 has 250 legislative seats to be apportioned among four states, where each state has a different population. Determine a population for each state in which Hamilton’s method, Jefferson’s method, Webster’s method, and Adams’ method all lead to the same apportionment of the 250 legislative seats. Many answers are possible.

When appropriate round quotas to the nearest hundredth.

Suppose that a police department has 210 new officers to apportion among six precincts. The department plans to apportion the officers based on the number of crimes committed during the previous year in each precinct. Suppose that the number of crimes committed in each precinct is different and that the total number of crimes committed in all six precincts was 2940. Determine the number of crimes committed in each precinct such that Hamilton’s method, Jefferson’s method, Webster’s method, and Adams’ method all lead to the same apportionment of the 210 new officers.

When appropriate round quotas to the nearest hundredth.

Fill in the blank with an appropriate word, phrase, or symbol(s).

When each group’s population is divided by the standard divisor, a standard _________ is obtained.

Fill in the blank with an appropriate word, phrase, or symbol(s).

The total population under consideration divided by the number of items to be allocated is called the standard ______.

The Sailing Club of Lakeport is holding an election to choose the club president. The 42 votes were cast as follows: Comstock, 20 votes; Owens, 15 votes; and Glazer, 7 votes.

(a) Using the plurality method, which candidate is elected president?

(b) Did this candidate receive a majority of votes?

Fill in the blank with an appropriate word, phrase, or symbol(s).

If a candidate receives a majority of first-place votes in an election, that candidate should be declared the winner. This criterion is called the _________ criterion.

Irvine Valley College is hiring a new director of counseling and advisement. The hiring committee ranks the four candidates, Alvarez (A), Brown (B), Singleton (S), and Yu (Y), according to the preference table below. Suppose that the Borda count method is used to determine the winner. Is the majority criterion satisfied? Explain.

Determine each hotel’s apportionment using Hamilton’s method.

When appropriate round quotas to the nearest hundredth.

A large hotel chain needs to apportion 25 new staff members among three hotels based on the numbers of rooms in each hotel as shown in the table below.

Construct a set of five pieces of data with a mean, median, mode, and midrange of 6 and a standard deviation of 0.

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