Question 1

A sailboat owner received 9 different signal flags with his new sailboat. If the order in which the flags are arranged on the mast determines the signal being sent, how many 4-flag signals can be sent? ​

Accepted Answer

To find the number of arrangements of 4 flags out of 9, we can use the formula for combinations: C(9,4) = 9!/(4!5!) = 126. However, since the order of the flags matters, we need to find the number of permutations of those 4 flags, which is 4! = 24. Therefore, the total number of 4-flag signals that can be sent is 126 x 24 = 3,024. The correct answer choice is E.

Question 2

A game of cards is played with a hand that consists of 3 cards dealt from a deck of 46 cards. How many different hands of cards are possible? ​

Accepted Answer

The number of different hands of 3 cards from a deck of 46 cards is calculated using combinations: C(46, 3) = 46! / (3!(46-3)!) = 15,180.

Question 3

In how many ways can a hand consisting of 4 spades, 3 hearts, 2 clubs, and 1 diamond be selected from a deck of 52 cards? ​

Accepted Answer

We can choose 4 spades out of 13 in 13 choose 4 ways. Similarly, we can choose 3 hearts out of 13 in 13 choose 3 ways, 2 clubs out of 13 in 13 choose 2 ways, and 1 diamond out of 13 in 13 choose 1 way. Therefore, the total number of ways to choose the hand is:

(13 choose 4) * (13 choose 3) * (13 choose 2) * (13 choose 1) = 207,352,860

Therefore, the best choice is C.

Question 4

An examination consists of 11 questions. If 6 questions must be answered, find the number of different orders in which a student can answer the questions. ​

Accepted Answer

This is a permutation problem since the order of answering the questions matters. The number of different orders can be found using the formula for permutation:

P(11,6) = 11!/(11-6)! = 11!/5! = 11 x 10 x 9 x 8 x 7 x 6 = 332,640

Therefore, the best choice is B.

Question 5

In a 10-question matching test with 10 possible answers to match and no answer used more than once, what is the probability of guessing and getting every answer correct? ​

Accepted Answer

The probability of guessing the first question correctly is 1/10. The probability of guessing the second question correctly is 1/9 (since one answer has already been used). This continues until the probability of guessing the last question correctly is 1/1. Therefore, the probability of guessing all 10 questions correctly is (1/10) x (1/9) x (1/8) x ... x (1/1) which simplifies to 1/10!. So, the probability is approximately 0.000008%. The closest answer choice to this is D.

Question 6

A senator asks his constituents to rank seven issues in order of importance. What is the probability that one reply chosen at random will rank the issues in alphabetical order? ​

Accepted Answer

The answer of A senator asks his constituents to rank...

Question 7

As a reward for a record year, the Ace Software Company is randomly selecting 5 people from its 70 employees for a free trip to Hawaii, but it will not pay for a traveling companion. If John and Jill are married and both are employees, what is the probability that they will both win? Round your answer to five decimal places. ​

Accepted Answer

The answer of As a reward for a record year,...

Question 8

Suppose an ice cream store offers 36 flavors of ice cream cones in sizes small, medium, and large. How many different selections of cones are possible if each cone has one flavor? ​

Accepted Answer

For each of the 36 flavors, there are 3 different sizes to choose from. Therefore, there are 3 options for each of the 36 flavors, resulting in a total of 3^36 possible cone selections. Using a calculator, this equals approximately 46,656.

Question 9

If 11 people are qualified for the next flight to a space station, how many different groups of 3 people can be chosen for the flight? ​

Accepted Answer

The number of ways to choose 3 people from 11 is calculated using combinations: C(11, 3) = 11! / (3!(11-3)!) = 165.

Question 10

In how many ways can a committee consisting of 3 men and 7 women be selected from a group consisting of 20 men and 17 women? ​

Accepted Answer

The answer of In how many ways can a committee...

Question 11

A department store manager wants to display 4 brands of a product along one shelf of an aisle. In how many ways can he arrange the brands? ​

Accepted Answer

The number of ways to arrange 4 items is 4! (4 factorial), which equals 24. Therefore, the manager can arrange 4 brands in 24 different ways.

Question 12

If the first digit of a seven-digit telephone number cannot be a 0 or 1, what is the probability that a number chosen at random will have all seven digits the same? ​

Accepted Answer

The answer of If the first digit of a seven-digit...

Question 13

Suppose an ice cream store offers 34 flavors of ice cream cones in sizes small, medium, and large. If four toppings are available to put on a cone, and a cone can be bought without a topping, how many different selections of ice cream are available if each cone has one flavor and one or no toppings? ​

Accepted Answer

There are three choices for sizes of cones, and each cone has 34 flavor options plus the option to have no flavor. Therefore, there are:

3 (cone size choices) x 35 (flavor and no flavor choices) = 105 options for just the cone and flavor.

For each of those 105 options, there are 5 choices for the toppings: no topping or any combination of the 4 toppings available.

Therefore, for each of the 105 cone and flavor options, there are 5 choices for toppings, giving us:

105 (cone and flavor options) x 5 (topping options) = 525 options for cone, flavor, and toppings.

Thus, the total number of different selections of ice cream available is:

105 (cone and flavor options) x 5 (topping options) = 525 options

Answer: C) 196,520.

Question 14

If a state wants each of its license plates to contain 6 different digits, how many different license plates can it make if any of the digits 0 through 9 can appear in any of the 6 positions? ​

Accepted Answer

For each of the 6 positions, there are 10 possible digits that can appear (0 through 9). Therefore, the total number of possible license plates is 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000.

Question 15

Five candidates for manager of a department store are ranked according to the weighted average of several criteria. How many different rankings are possible for the top four candidates if no two candidates receive the same rank? ​

Accepted Answer

There are 5 candidates and 4 positions to be ranked. The first position can be filled by any of the 5 candidates, the second position can be filled by any of the remaining 4 candidates, the third by any of the remaining 3 candidates, and the fourth by either of the remaining 2 candidates. Therefore, the total number of possible rankings is 5 x 4 x 3 x 2 = 120.

Question 16

A senator asks his constituents to rank six issues in order of importance. How many rankings are possible? ​

Accepted Answer

The number of possible rankings can be found using factorials. In this case, there are 6 choices for the first ranking, 5 for the second, 4 for the third, and so on, until only one choice is left for the last ranking. Therefore, the number of possible rankings is 6 x 5 x 4 x 3 x 2 x 1 = 720, which is option D.

Question 17

Suppose that all license plates in a state have three of the letters A through Z followed by four of the digits 0 through 9. If a plate is chosen at random, what is the probability that all three letters and all four digits on the plate will be different? Round your answer to five decimal places. ​

Accepted Answer

There are 26 choices for each letter and 10 choices for each digit. Therefore, there are $26^3\cdot 10^4$ possible license plates. To count the number of license plates with all three letters and all four digits different, we use the multiplication principle. For the letters, there are 26 choices for the first letter, 25 choices for the second letter (since the first letter has already been chosen), and 24 choices for the third letter (since two have already been chosen). For the digits, there are 10 choices for the first digit, 9 choices for the second digit (since the first digit has already been chosen), 8 choices for the third digit (since two have already been chosen), and 7 choices for the fourth digit (since three have already been chosen). Therefore, there are $26\cdot 25\cdot 24\cdot 10\cdot 9\cdot 8\cdot 7$ license plates with all three letters and all four digits different. The probability of choosing one of these license plates at random is:
$$\frac{26\cdot 25\cdot 24\cdot 10\cdot 9\cdot 8\cdot 7}{26^3\cdot 10^4} = 0.447337$$
Therefore, the answer is (C) 0.447337.

Question 18

To determine voters' feelings regarding an issue, a candidate asks a sample of people to pick 5 words out of a set of 6 words that they feel best describe the issue. How many different groupings of 5 words are possible? ​

Accepted Answer

Since there are 6 words and each person has to choose 5 out of those, it's essentially the same as excluding 1 word out of 6. This can be done in 6 different ways, hence the answer is 6.

Question 19

An automatic teller machine requires that each customer enter a three-digit personal identification number (PIN) when he or she inserts a bank card. If a person finds a bank card and guesses at a PIN to use the card fraudulently, what is the probability that the person will succeed in one attempt? ​

Accepted Answer

The answer of An automatic teller machine requires that each...

Question 20

An automatic teller machine requires that each customer enter a five-digit personal identification number (PIN) when he or she inserts a bank card. If a person finds a bank card and guesses at a PIN to use the card fraudulently, and the person knows that the PIN will not have any digit repeated, what is the probability the person will succeed in guessing in one attempt? ​

Accepted Answer

The answer of An automatic teller machine requires that each...

Quiz 8: Introduction to Probability

A sailboat owner received 9 different signal flags with his new sailboat. If the order in which the flags are arranged on the mast determines the signal being sent, how many 4-flag signals can be sent? ​

A game of cards is played with a hand that consists of 3 cards dealt from a deck of 46 cards. How many different hands of cards are possible? ​

In how many ways can a hand consisting of 4 spades, 3 hearts, 2 clubs, and 1 diamond be selected from a deck of 52 cards? ​

An examination consists of 11 questions. If 6 questions must be answered, find the number of different orders in which a student can answer the questions. ​

In a 10-question matching test with 10 possible answers to match and no answer used more than once, what is the probability of guessing and getting every answer correct? ​

A senator asks his constituents to rank seven issues in order of importance. What is the probability that one reply chosen at random will rank the issues in alphabetical order? ​

Suppose an ice cream store offers 36 flavors of ice cream cones in sizes small, medium, and large. How many different selections of cones are possible if each cone has one flavor? ​

If 11 people are qualified for the next flight to a space station, how many different groups of 3 people can be chosen for the flight? ​

In how many ways can a committee consisting of 3 men and 7 women be selected from a group consisting of 20 men and 17 women? ​

A department store manager wants to display 4 brands of a product along one shelf of an aisle. In how many ways can he arrange the brands? ​

If the first digit of a seven-digit telephone number cannot be a 0 or 1, what is the probability that a number chosen at random will have all seven digits the same? ​

If a state wants each of its license plates to contain 6 different digits, how many different license plates can it make if any of the digits 0 through 9 can appear in any of the 6 positions? ​

Five candidates for manager of a department store are ranked according to the weighted average of several criteria. How many different rankings are possible for the top four candidates if no two candidates receive the same rank? ​

A senator asks his constituents to rank six issues in order of importance. How many rankings are possible? ​

To determine voters' feelings regarding an issue, a candidate asks a sample of people to pick 5 words out of a set of 6 words that they feel best describe the issue. How many different groupings of 5 words are possible? ​

A sailboat owner received 9 different signal flags with his new sailboat. If the order in which the flags are arranged on the mast determines the signal being sent, how many 4-flag signals can be sent?

A game of cards is played with a hand that consists of 3 cards dealt from a deck of 46 cards. How many different hands of cards are possible?

In how many ways can a hand consisting of 4 spades, 3 hearts, 2 clubs, and 1 diamond be selected from a deck of 52 cards?

An examination consists of 11 questions. If 6 questions must be answered, find the number of different orders in which a student can answer the questions.

In a 10-question matching test with 10 possible answers to match and no answer used more than once, what is the probability of guessing and getting every answer correct?

A senator asks his constituents to rank seven issues in order of importance. What is the probability that one reply chosen at random will rank the issues in alphabetical order?

Suppose an ice cream store offers 36 flavors of ice cream cones in sizes small, medium, and large. How many different selections of cones are possible if each cone has one flavor?

If 11 people are qualified for the next flight to a space station, how many different groups of 3 people can be chosen for the flight?

In how many ways can a committee consisting of 3 men and 7 women be selected from a group consisting of 20 men and 17 women?

A department store manager wants to display 4 brands of a product along one shelf of an aisle. In how many ways can he arrange the brands?

If the first digit of a seven-digit telephone number cannot be a 0 or 1, what is the probability that a number chosen at random will have all seven digits the same?

If a state wants each of its license plates to contain 6 different digits, how many different license plates can it make if any of the digits 0 through 9 can appear in any of the 6 positions?

Five candidates for manager of a department store are ranked according to the weighted average of several criteria. How many different rankings are possible for the top four candidates if no two candidates receive the same rank?

A senator asks his constituents to rank six issues in order of importance. How many rankings are possible?

To determine voters' feelings regarding an issue, a candidate asks a sample of people to pick 5 words out of a set of 6 words that they feel best describe the issue. How many different groupings of 5 words are possible?