Solve the problem using inductive reasoning. 1) In how many ways can you exactly cover the last two diagrams with "dominoes" that are just the size of two small squares?
2) Find the number of games played in a round robin tournament for the given numbers of teams. In a round robin tournament every team plays every other team once. Number of teams Number of games played 3 teams 3 games 4 teams 6 games 5 teams 10 games 6 teams __ games 7 teams __ games Look for a pattern. Find the number of games played in a round robin tournament involving n teams. Find the number of games played in a round robin tournament involving 16 teams. Draw the next figure in the pattern. 3)
4)
5)
1
Use inductive reasoning to predict the next line in the pattern. 6) 9 × 9 = 81 99 × 99 = 9801 999 × 999 = 998,001 7) 40 - 9 = 31 400 - 89 = 311 4000 - 789 = 3211 Use inductive reasoning to predict the next number in the sequence. 8) 0, 4, 4, 0, -4, ... 9) 3, 5, 6, 10, 12, 20, ... Solve the problem using inductive reasoning. 10) Find the next term in the following sequence. F, S, S, M, T 11) Find the 4th triangular number that corresponds to the following dot sequence.
Estimate the answer from the table or graph. 12) The number of students at Alder High School who studied foreign languages in different years is shown in the bar graph. What is the total number of students who studied a foreign language in 2012? (Assume no student studied two foreign languages).
2006
2008 2010 Year A) 150 students
2012 B) 130 students
C) 90 students
2
D) 170 students
13) In a shop that sells a variety of nuts, the prices of some items are as given below. If Sarah buys 2 lb of cashews, 1 lb of walnuts, and 2 lb of raisins, how much did she have to pay? Item Cost/lb Almonds $4.30 Walnuts $3.80 Cashews $4.80 Pecans $3.80 Raisins $3.50
14) The graph shows the average monthly cost of a wireless phone service for the years 2005 through 2012. Estimate the average monthly cost of this wireless phone service in 2006. 70
y
60
Cost (dollars)
50 40 30 20 10
2005 2006 2007 2008 2009 2010 2011 2012
x
Year
A) $37
B) $44
C) $34
D) $31
Solve the problem. 15) One gallon of a driveway sealant covers an area of 180 ft2. How many gallons of the sealant are needed to cover a 900 ft2 driveway? 16) To make orange juice from concentrate powder, you need to mix 2.5 teaspoons of the concentrate in 16 ounces of water. How much concentrate powder do you need for 1 gallon of water? 17) The cost of gasoline is $4.40 per gallon. Jane's car gives a mileage of 35 miles per gallon. Approximately how much did Jane pay for gasoline for a trip of 491 miles? 18) An airport parking lot charges $4.50 for the first two hours of parking and $1.00 for each additional half hour or part thereof. If Sam parks his car for 7 hours, how much does he pay for parking? 19) A small farm field is a square measuring 350 ft on a side. What is the perimeter of the field? If you double the length of each side of the field, what is the new perimeter? 20) A boxer takes 3 drinks of water between each round for the first four rounds of a championship fight. After the fourth round he starts to take his three drinks plus one additional drink between each of the remaining rounds. If he continues to increase his drinks by 1 after each round, how many drinks will he take between the 14th and 15th round?
3
21) Missy and Adam work at different jobs. Missy earns $7 per hour and Adam earns $5 per hour. They each earn the same amount per week but Adam works 2 more hours. How many hours a week does Adam work? 22) An average newspaper contains at least 16 pages and at most 87 pages. How many newspapers must be collected to be certain that at least two newspapers have the same number of pages? Use the table or graph to answer the question. 23) Amy graphed her utility bills for the last year for her records. Estimate the total amount Amy paid for her utilities for the month of January.
24) The following chart shows an appliance store's average percent profit margin on certain items: Product category Average profit margin,% Washer/Dryer 17 Refrigerator 13 Stove 16 Microwave 40 What is the average profit for the store if it lists the price of a particular refrigerator at $800? Complete the magic (addition) square. 25) Use each number 26, 27, 28, 29, 30, 31, 32, 33, and 34 once. 31 30 28 27 34
26) Use each number 20, 21, 22, 23, 24, 25, 26, 27, and 28 once. 22 23 24 28 21
4
Solve the problem. 27) A die is rolled 50 times with the following results. Outcome Frequency
1 2 3 4 5 6 11 2 24 8 0 5
Compute the empirical probability that the die comes up a 5. 28) Three coins are tossed 80 times and the number of heads is observed. Outcome Frequency
no heads one head two heads three heads 9 15 33 23
Compute the empirical probability that at most two heads occur. 29) This spinner is spun 36 times. The spinner landed on A 17 times, on B 11 times, and on C 8 times. Compute the empirical probability that the spinner will land on B.
30) A die is rolled 100 times with the following results. Outcome Frequency
1 2 3 4 5 6 12 12 28 28 11 9
Compute the empirical probability that the die comes up 2 or 3. 31) Two coins are tossed 20 times and the number of tails is observed. Outcome Frequency
2 tails 1 tail 0 tails 6 8 6
Compute the empirical probability that exactly one tail occurred. Find the probability of the following five-card poker hands from a 52-card deck. In poker, aces are either high or low. 32) Four of a kind (4 cards of the same value) 33) Full house (3 cards of one value, 2 of another value) 34) Straight (5 in a row, but not a straight flush) 35) Flush (5 in same suit, but not a straight flush) 36) Royal flush (5 highest cards of a single suit)
5
Find the probability. Round to the nearest ten-thousandth when necessary. 37) A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. Find the probability that you have all cherry candies. 38) A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. Find the probability that you have 1 cherry candy and 2 lemon candies. 39) A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. What is the probability that you have at least 2 cherry candies? 40) A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. Find the probability that you have 2 orange candies and 1 lemon candy. 41) A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. What is the probability that you have at least 2 orange candies? 42) A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. What is the probability that you have at least 1 lemon candy? 43) A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. Find the probability that you have all lemon candies. 44) A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. Find the probability that you have one candy of each flavor. 45) A coin is biased to show 42% heads and 58% tails. The coin is tossed twice. What is the probability that the coin turns up heads on the second toss? 46) A coin is biased to show 39% heads and 61% tails. The coin is tossed twice. What is the probability that the coin turns up heads once and tails once? 47) A fair coin is tossed 5 times. What is the probability of exactly 2 head(s)? Find the probability. 48) A child rolls a 6-sided die 6 times. What is the probability of the child rolling exactly four fives? Round to the nearest ten-thousandth. 49) A child rolls a 6-sided die 6 times. What is the probability of the child rolling exactly three sixes? Round to the nearest ten-thousandth. 50) A child rolls a 6-sided die 6 times. What is the probability of the child rolling no more than three twos? Round to the nearest ten-thousandth. 51) A die is rolled 20 times and the number of twos that come up is tallied. Find the probability of getting more than three twos. Round to the nearest thousandth when necessary. 52) In a certain college, 33% of the physics majors are ethnic minorities. A random sample of 10 physics majors is chosen. Find the probability that 2 or less are ethnic minorities. Round to the nearest ten-thousandth.
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53) In a certain college, 33% of the physics majors are ethnic minorities. A random sample of 10 physics majors is chosen. Find the probability that 7 or more are ethnic minorities. Round to the nearest ten-thousandth. 54) In a certain college, 33% of the physics majors are ethnic minorities. A random sample of 10 physics majors is chosen. Find the probability that no more than 6 are ethnic minorities. Round to the nearest ten-thousandth. 55) In a certain college, 33% of the physics majors are ethnic minorities. A random sample of 10 physics majors is chosen. Find the probability that exactly 2 are ethnic minorities. Round to the nearest ten-thousandth. 56) On a hospital floor, 16 patients have a disease with a mortality rate of 0.1. Two of them die. Round to the nearest thousandth. 57) The probability that a radish seed will germinate is 0.7. The gardener plants 20 seeds and she harvests 16 radishes. Round to the nearest thousandth. 58) A die is rolled 18 times and two threes come up. Round to the nearest thousandth. 59) If a person is randomly selected, find the probability that his or her birthday is in May. Ignore leap years. Assume that all days of the year are equally likely for a given birth. 60) Determine the probability that the spinner lands on grey.
61) A bag contains 7 red marbles, 9 blue marbles, and 6 green marbles. What is the probability of choosing a blue marble? 62) Determine the probability that the spinner lands on white.
7
63) Determine the probability that the spinner lands on white.
64) Determine the probability that the spinner lands on white.
65) A lottery game has balls numbered 1 through 15. What is the probability of selecting an even numbered ball or a 8? 66) When two balanced dice are rolled, there are 36 possible outcomes. Find the probability that the sum is a multiple of 3 or greater than 4. 67) One card is selected from a deck of cards. Find the probability that the card selected is greater than 3 and less than 8. 68) A spinner has regions numbered 1 through 21. What is the probability that the spinner will stop on an even number or a multiple of 3? 69) One digit from the number 1,838,228 is written on each of seven cards. What is the probability of drawing a card that shows 1 or 2? 70) When two balanced dice are rolled, there are 36 possible outcomes. Find the probability that either doubles are rolled or the sum of the dice is 8. 71) A card is drawn at random from a standard 52-card deck. Find the probability that the card is neither an ace nor a heart. 72) Two fair dice are rolled. Find the probability that the sum of the two numbers is not greater than 5. 73) If a person is randomly selected, find the probability that his or her birthday is not in May. Ignore leap years.
8
74) The distribution of B.A. degrees conferred by a local college is listed below, by major. Major Number of Students English 2073 Mathematics 2164 Chemistry 318 Physics 856 Liberal Arts 1358 Business 1676 Engineering 868 Total:
9313
What is the probability that a randomly selected degree is not in Mathematics? Round to the nearest thousandth. 75) The chart below gives the cost and number of vehicle tags sold in each city. City Cost of Vehicle Tag Number of Vehicle Tags Sold Bristol $8 1961 Trevor $10 3260 Camp Lake $7 2538 Salem $15 1610 Paddock Lake $12 2,541 One car is selected at random from the cars with vehicle tags from these cities. What is the probability that this car has a vehicle tag that cost less than $10? Round to the nearest ten-thousandth when necessary. 76) A card is drawn at random from a standard 52-card deck. Find the probability that the card is an ace or not a club. Find the odds. 77) A number cube labeled with numbers 1, 2, 3, 4, 5, and 6 is tossed. What are the odds in favor of the cube showing a 4? 78) What are the odds in favor of drawing a 3 from these cards? 79) A number cube labeled with numbers 1, 2, 3, 4, 5, and 6 is tossed. What are the odds in favor of the cube showing an odd number? 80)
What are the odds in favor of spinning a D on this spinner?
9
81)
What are the odds in favor of spinning an A on this spinner? 82) What are the odds in favor of drawing an even number from these cards? Solve the problem. 83) The odds in favor of Trudy beating her friend in a round of golf are 1 : 6. Find the probability that Trudy will lose. 84) The odds in favor of Carl beating his friend in a round of golf are 5 : 4. Find the probability that Carl will lose. 85) The odds in favor of Carl beating his friend in a round of golf are 7 : 2. Find the probability that Carl will beat his friend. 86) The odds in favor of a horse winning a race are posted as 6 : 5. Find the probability that the horse will lose the race. 87) If the probability that an identified hurricane will make a direct hit on a certain stretch of beach is 0.25, what are the odds in favor of a direct hit? 88) If it has been determined that the probability of an earthquake occurring on a certain day in a certain area is 0.01, what are the odds against an earthquake? 89) The chart shows winnings, in dollars, for the 10 highest-rated FASTCAR drivers for last driving season. Driver Rick Bobby Maverick St. Joseph Johnny Wright Jay Smith William Rock Bob Ricky Jimmy Novak Tommy Keefe Tyler Jones
Winnings $6,394,185 $5,198,600 $2,140,478 $4,773,112 $2,951,951 $4,020,078 $4,420,069 $5,501,450 $3,127,812
If one of the drivers listed in the chart is selected at random, determine the odds against the driver earning more than $4 million last season. 90) In a certain town, 20% of people commute to work by bicycle. If a person is selected randomly from the town, what are the odds against selecting someone who commutes by bicycle?
10
91) A contractor is considering a sale that promises a profit of $39,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such) of $14,000 with a probability of 0.3. What is the expected profit? 92) From a group of 3 men and 4 women, a delegation of 2 is selected. What is the expected number of men in the delegation? 93) At age 50, Ann must choose between taking $21,000 at age 60 if she is alive then, or $31,000 at age 70 if she is alive then. The probability for a person aged 50 living to be 60 and 70 is 0.81 and 0.61, respectively. Using expected value, what is Ann's best option? 94) An insurance company says that at age 50 one must choose to take $10,000 at age 60, $30,000 at 70, or $50,000 at 80 ($0 death benefit). The probability of living from 50 to 60 is 0.84, from 50 to 70, 0.64, and from 50 to 80, 0.44. Find the expected value at each age. 95) Find the expected number of girls in a family of 6 children. 96) An insurance company has written 52 policies of $50,000, 477 of $25,000, and 918 of $10,000 on people of age 20. If the probability that a person will die at age 20 is 0.001, how much can the company expect to pay during the year the policies were written? 97) Find the expected number of boys in a family of 4 children. 98) An insurance company will insure a $240,000 home for its total value for an annual premium of $570. If the company spends $30 per year to service such a policy, the probability of total loss for such a home in a given year is 0.001 and you assume either total loss or no loss will occur, what is the company's expected annual gain (or profit) on each such policy? 99) Find the expected number of girls in a family of 3 children. 100) If 2 cards are drawn from a deck of 52 cards, what is the expected number of spades? 101) A company estimates that it has a 40% chance of being successful in bidding on a $35,000 contract. If it costs $6000 in consultant fees to prepare the bid, what is the expected gain or loss for the company if it decides to bid on this contract? 102) Assume that you have a car worth $6100 and you wish to insure it for its full replacement value if it is stolen. If there is a 1% chance that the car will be stolen, what would a fair premium price be? 103) In a game, you have a 1/39 probability of winning $105 and a 38/39 probability of losing $3. What is your expected winning? 104) Suppose you pay $3.00 to roll a fair die with the understanding that you will get back $9.00 for rolling a 6 or a 1, nothing otherwise. What are your expected winnings? 105) Suppose you pay $1.00 to roll a fair die with the understanding that you will get back $3.00 for rolling a 4 or a 6, nothing otherwise. What is your expected net winnings?
11
106) Bob and Fred play the following game. Bob rolls a single die. If an even number results, Bob must pay Fred the number of dollars indicated by the number rolled. On the other hand, if an odd number is rolled, Fred must pay Bob the number of dollars indicated by the number rolled. Find Bob's expectation. 107) Bob and Fred play the following game. Bob rolls a single die. If the result is the number 1, 2, 3, 4, or 5 , Bob must pay Fred the number of dollars indicated by the number rolled. If a 6 is rolled, Fred must pay Bob $45. Find Fred's expectation. 108) Suppose you buy 1 ticket for $1 out of a lottery of 1,000 tickets where the prize for the one winning ticket is to be $500. What are your expected winnings? 109) In a raffle, one thousand tickets are sold. There is a grand prize of $6000, a second prize of $400, and a third prize of $100. What ticket price would make this game fair? 110) Assume that a person spins the pointer and is awarded the amount indicated if the pointer points to a positive number but must pay the amount indicated if the pointer points to a negative number. Determine the person's expectation.
$9 -$7 $1
Use the counting principle to obtain the answer. 111) A saleswoman packed 3 jackets and 5 skirts. With one jacket, she could wear all 5 skirts. With another jacket, she could wear 4 skirts. With the third jacket, she could wear only 3 skirts. How many different combinations did she have? 112) A salesman packed 3 shirts and 5 ties. With one shirt, he could wear all 5 ties. With another shirt, he could wear 4 ties. With the third shirt, he could wear only 2 ties. How many different combinations did he have? 113) A local department store sold carpets in 4 sizes. Each carpet came in 3 qualities. One size of carpet came in 7 colors. The other sizes came in 4 colors. How many choices of carpet were there? 114) The frequency setting for a garage door opener is determined by the positions of six switches, each of which can be set to a "+" or "-" position. In how many ways can the switches be set? 115) A restaurant offered salads with 2 types of lettuce, 4 different toppings, and 3 different dressings. How many different types of salad could be offered? 116) A sports shop sold tennis rackets in 2 different weights, 3 types of string, and 4 grip sizes. How many different rackets could they sell? 12
117) A restaurant offered salads with 6 types of dressings and 4 different toppings. How many different types of salads could be offered? Solve the problem. 118) There are 3 cards in a hat; one is a king, one is a queen, and one is an ace. Two cards are to be selected at random with replacement. Using a tree diagram , obtain the sample space for the experiment. Then, find the probability that you choose the same card twice. 119) There are 3 cards in a hat; one is a king, one is a queen, and one is an ace. Two cards are to be selected at random without replacement. Using a tree diagram , obtain the sample space for the experiment. Then, find the probability that a king and a queen are selected. 120) There are 3 cards in a hat; one is a king, one is a queen, and one is an ace. Two cards are to be selected at random with replacement. Using a tree diagram , obtain the sample space for the experiment. Then, find the probability that a king and a queen are selected. 121) There are 3 balls in a hat; one with the number 1 on it, one with the number 5 on it, and one with the number 7 on it. You pick a ball from the hat at random and then you flip a coin. Using a tree diagram , obtain the sample space for the experiment. Then, find the probability that the number on the ball is greater than 6. 122) There are 3 cards in a hat; one is a king, one is a queen, and one is an ace. Two cards are to be selected at random without replacement. Using a tree diagram , obtain the sample space for the experiment. Then, find the probability that you choose the same card twice. 123) There are 3 balls in a hat; one with the number 1 on it, one with the number 2 on it, and one with the number 5 on it. You pick a ball from the hat at random and then you roll a die. Using a tree diagram , obtain the sample space for the experiment. Then, find the probability that the number on the ball is greater than the number on the die. 124) A couple plans to have four children. Using a tree diagram , obtain the sample space. Then, find the probability that the family has no boys. 125) Use a tree diagram showing all possible results when four fair coins are tossed to list the ways of getting exactly two tails. 126) A couple plans to have four children. Using a tree diagram , obtain the sample space. List the elements that make up the sample space. (Use "B" for "boy" and "G" for "girl.") 127) There are 3 balls in a hat; one with the number 3 on it, one with the number 4 on it, and one with the number 9 on it. You pick a ball from the hat at random and then you flip a coin. Using a tree diagram , obtain the sample space for the experiment. List the elements that make up the sample space. 128) Use a tree diagram showing all possible results when a die is rolled twice to list the ways of getting exactly one die showing a 3. 129) Use a tree diagram showing all possible results when a die is rolled twice to list the ways of getting at least one die showing a 3.
13
130) There are 3 cards in a hat; one is a King, one is a Queen, and one is an Ace. Two cards are to be selected at random without replacement. Using a tree diagram , obtain the sample space for the experiment. List the elements that make up the sample space. 131) There are 3 balls in a hat; one with the number 1 on it, one with the number 2 on it, and one with the number 5 on it. You pick a ball from the hat at random and then you roll a die. Using a tree diagram , obtain the sample space for the experiment. List the elements that make up the sample space. Find the probability. 132) Elise has put 5 cans (all of the same size) on her kitchen counter; 2 cans of vegetables, 2 cans of soup , and 1 can of peaches. Her son, Ryan, takes the labels off the cans and throws them away. Elise then chooses 2 cans (without replacement) at random to open. Find the probability that she will open at least 1 can of soup. 133) You are dealt two cards successively (without replacement) from a shuffled deck of 52 playing cards. Find the probability that the first card is a king and the second card is a queen. 134) A basketball player hits three-point shots 43% of the time. If she takes 4 shots during a game, what is the probability that she misses the first shot and hits the last three shots? Round to the nearest tenth of a percent when necessary. 135) If you are dealt two cards successively (with replacement of the first) from a standard 52-card deck, find the probability of getting a heart on the first card and a diamond on the second. 136) A basketball player hits three-point shots 41% of the time. If she takes 4 shots during a game, what is the probability that she hits all 4 shots? Round to the nearest tenth of a percent when necessary. 137) Find the probability of correctly answering the first 3 questions on a multiple choice test if random guesses are made and each question has 5 possible answers. 138) A lottery game has balls numbered 0 through 13. What is the probability of selecting an even numbered ball or a 7? 139) Each of ten tickets is marked with a different number from 1 to 10 and put in a box. If you draw a ticket from the box, what is the probability that you will draw 7, 8, or 1? Solve the problem. 140) A single die is rolled one time. Find the probability of rolling an odd number or a number less than 5. 141) One card is selected from a deck of cards. Find the probability of selecting a diamond or a card less than 7. (Note: The ace is considered a low card.) 142) One card is selected from a deck of cards. Find the probability of selecting a red card or a heart . 143) Of the 58 people who answered "yes" to a question, 15 were male. Of the 63 people who answered "no" to the question, 9 were male. If one person is selected at random from the group, what is the probability that the person answered "yes" or was male? Round to the nearest thousandth when necessary.
14
144) A survey of senior citizens at a doctor's office shows that 47% take blood pressure-lowering medication, 45% take cholesterol-lowering medication, and 8% take both medications. What is the probability that a senior citizen takes either blood pressure-lowering or cholesterol-lowering medication? Round to the nearest hundredth. 145) A single die is rolled one time. Find the probability of rolling a number greater than 2 or less than 6. 146) One card is selected from a deck of cards. Find the probability of selecting a black card or a card less than 8. (Note: The ace is considered a low card.) Find the indicated probability. 147) If P(A) = 0.3, P(B) = 0.3, and P(A and B) = 0.3, find P(A or B). 148) If P(B) = 0.3, P(A or B) = 0.4, and P(A and B) = 0.5, find P(A). Solve the problem. 149) If two cards are drawn without replacement from a deck, find the probability that the second card is a face card, given that the first card was a queen. 150) If a single fair die is rolled, find the probability of a 4 given that the number rolled is odd. 151) If two fair dice are rolled, find the probability of a sum of 6 given that the roll is a "double". 152) If a single fair die is rolled, find the probability of a 5 given that the number rolled is odd. 153) If three cards are drawn without replacement from a deck, find the probability that the third card is a face card, given that the first card was a queen and the second card was a 5. 154) If two cards are drawn without replacement from a deck, find the probability that the second card is a spade, given that the first card was a spade. Two marbles are drawn without replacement from a box with 3 white, 2 green, 2 red, and 1 blue marble. Find the probability. 155) The second marble is blue given the first marble is white. 156) The second marble is blue given the first marble is red. 157) One marble is white and one marble is blue. 158) The second marble is white given the first marble is blue. 159) Both marbles are green. 160) One marble is green and one marble is red.
15
Use the table to find the probability. 161) The following table contains data from a study of two airlines which fly to Small Town, USA. Number of flights
Number of flights
arrived on time
arrived late
Podunk Airlines
33
6
Upstate Airlines 43 5 If a flight is selected at random, find the probability that the flight is on Upstate Airline given the flight is late. 162) The following table contains data from a study of two airlines which fly to Small Town, USA. Number of flights
Number of flights
arrived on time
arrived late
Podunk Airlines
33
6
Upstate Airlines 43 5 If a flight is selected at random, find the probability that the flight will arrive on time. 163) The following table indicates the preference for different types of soft drinks by three age groups. under 21 years of age between 21 and 40 over 40 years of age
cola root beer lemon-lime 40 25 20 35 20
20 30
30 35
If a person is selected at random, find the probability that the person drinks root beer given that they are over 40. 164) The following table contains data from a study of two airlines which fly to Small Town, USA.
Podunk Airlines
Number of flights
Number of flights
arrived on time
arrived late
33
6
Upstate Airlines 43 5 If a flight is selected at random, find the probability that the flight will arrive on time and that they fly with Upstate airlines. 165) The following table contains data from a study of two airlines which fly to Small Town, USA.
Podunk Airlines
Number of flights
Number of flights
arrived on time
arrived late
33
6
Upstate Airlines 43 5 If a flight is selected at random, find the probability that the flight will arrive on time given that they fly with Upstate airlines.
16
166) The following table indicates the preference for different types of soft drinks by three age groups. cola root beer lemon-lime under 21 years of age 40 25 20 between 21 and 40 over 40 years of age
35 20
20 30
30 35
If a person is selected at random, find the probability that the person is over 40 years of age given that they drink root beer. An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul. 167) In how many ways can the first award be presented to Karen and the last to Lyle? 168) In how many ways can the men be presented first and then the women? 169) In how many ways can the people be presented? Suppose a traveler wanted to visit a museum, an art gallery, and the state capitol building. 45-minute tours are offered at each attraction hourly from 10 a.m. through 3 p.m. (6 different hours). Solve the problem, disregarding travel time. 170) In how many ways could the traveler schedule all three tours before 1 p.m.? 171) In how many ways can the traveler visit all three places in one day? 172) In how many ways could the traveler schedule all three tours in one day, with the art gallery being the last tour of the day? 173) In how many ways could the traveler schedule two of the three tours in one day? Solve the problem. 174) How many different three-digit numbers can be written using digits from the set {2, 3, 4, 5, 6} without any repeating digits? 175) How many ways can the letters in the word "WISCONSIN" be arranged? 176) A license plate is to consist of 2 letters followed by 4 digits. Determine the number of different license plates possible if repetition of letters and numbers is permitted. 177) A license plate is to consist of 3 letters followed by 5 digits. Determine the number of different license plates possible if the first letter must be an N , M , or P and repetition of letters and numbers is not permitted. 178) In how many ways can the letters in the word PAYMENT be arranged if the letters are taken 4 at a time? 179) A signal is made by placing 3 flags, one above the other, on a flag pole. If there are 7 different flags available, how many possible signals can be flown? Evaluate the expression. 180) 10P5
17
181) 6 P0 Solve the problem. 182) License plates are made using 2 letters followed by 2 digits. How many plates can be made if repetition of letters and digits is allowed? 183) How many three-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, if repetitions of digits are allowed? 184) How many different sequences of 4 digits are possible if the first digit must be 3, 4, or 5 and if the sequence may not end in 000? Repetition of digits is allowed. 185) How many different 4-letter radio-station call letters can be made if the first letter must be K or W, repeats are allowed, but the call letters cannot end in an O? 186) How many ways can a committee of 6 be selected from a club with 10 members? 187) Three noncollinear points determine a triangle. How many triangles can be formed with 8 points, no three of which are collinear? 188) In how many ways can a student work 7 out of 10 questions on an exam? 189) In how many ways can a group of 7 students be selected from 8 students? 190) Bob is planning to pack 6 shirts and 4 pairs of pants for a trip. If he has 8 shirts and 7 pairs of pants to choose from, in how many different ways can this be done? 191) A bag contains 6 apples and 4 oranges. If you select 5 pieces of fruit without looking, how many ways can you get exactly 4 apples? 192) A bag contains 7 apples and 5 oranges. If you select 6 pieces of fruit without looking, how many ways can you get 6 oranges? 193) If you toss six fair coins, in how many ways can you obtain at least two heads? 194) The chorus has six sopranos and eight baritones. In how many ways can the director choose a quartet that contains at least one soprano? 195) If you toss five fair coins, in how many ways can you obtain at least one head? Evaluate the expression. 196)
8 P4 11C7
18
197)
9 C4 9 C3
198) 8 C4 Give an appropriate answer. 199) Just prior to a gubernatorial election, a survey of 1000 likely voters is conducted to determine the level of support for the Democrat and Republican candidates for governor. The results are as follows: 38% favor the Democrat candidate, 23% prefer the Republican candidate, and 39% are undecided. Based on these results, and assuming that the survey was unbiased, would you feel comfortable concluding that the Democrat candidate will win the election? Explain. 200) A statistics student takes a poll on her campus to find out which presidential candidate the sample of students favors. Is she employing descriptive or inferential statistics? 201) A statistics student takes a poll of residents at a local retirement home to find out which presidential candidate the sample of residents favors. Based on her results, she makes a prediction about which candidate will win the election. Comment on the fairness of her prediction. What might she do to improve her poll? Identify the sampling technique used to obtain a sample. 202) Of all the fishermen on a lake, the game warden chooses to check the fishing license only of the first person that returns to the boat dock. 203) A group of people are classified according to height and then random samples of people from each group are taken. 204) A game warden chooses to check for fishing licenses in every 7th boat he encounters on his patrol. Tell what possible misuses or misinterpretations may exist for the following statement. 205) More adults than teenagers are involved in automobile accidents each year. Therefore, teenagers are better drivers than adults. 206) The average yearly income in a certain country is $39,000. Therefore, everyone in this country is financially secure. 207) More people drown at ocean beaches each year than at lake beaches. Therefore, ocean beaches are more dangerous.
19
Give an appropriate answer. 208) The following table shows the average life expectancy in a certain country in the given years. Year 1990 1995 2000 2005 2010
Life Expectancy (years) 72.6 73.7 74.7 75.4 75.8
Draw a bar graph that makes the increase in the life expectancy look large.
209) The following table shows the average income of families in a certain state from 2008 through 2012. Year 2008 2009 2010 2011 2012
Average Income ($) $37,000 $39,000 $41,000 $42,000 $45,000
Draw a bar graph that makes the increase look small.
20
210) The following table shows the average price per pound of seed from 2008 through 2012. Year 2008 2009 2010 2011 2012
Average price per lb ($) $0.78 $0.72 $0.83 $0.86 $0.90
Draw a bar graph that makes the increase in the price look large.
Use the data to construct a frequency distribution. 211) 34 34 34 35 35 35 35 36 36 38 First Class = 34-36 38 38 39 39 39 40 40 41 41 42 212) 43 44 44 45 48 48 48 49 49 49 49 50 50 51 51
First Class = 43-44
213) A car insurance company conducted a survey to find out how many car accidents people had been involved in. They selected a sample of 32 adults between the ages of 30 and 70 and asked each person how many accidents they had been involved in the past ten years. The following data were obtained. 0 1 2 1
1 1 0 3
0 1 0 0
3 0 1 0
2 2 0 1
1 0 2 0
0 4 1 5
2 1 3 4
Construct a frequency distribution, letting each class have a width of one.
21
214) A teacher asked each of her students how many novels they had read in the previous six months. The results are shown below. 0 2 1 7 0
1 7 2 1 2
5 2 6 4 1
4 5 0 2 1
2 0 2 3 0
1 1 3 1 6
3 0 1 7 1
2 1 2 0 7
Construct a frequency distribution, letting each class have a width of one. Construct a frequency polygon. 215) Weight Number of (in pounds) students 101-130 14 131-160 12 161-190 68 191-220 76 221-250 30
216) Life of bulb Number (in hours) of bulbs 400-499 45 500-599 80 600-699 120 700-799 70 800-899 35
22
217) Height Number of (in inches) employees 60-63 26 64-67 63 68-71 49 72-75 38 76-79 24
218) Weight of bag Number of of chips (in ounces) bags 15.6-15.7 6 15.8-15.9 22 16.0-16.1 35 16.2-16.3 27 16.4-16.5 10
23
Construct a histogram of the given frequency distribution. 219) The frequency distribution indicates the age of 2190 students in a college chemistry course. Age (years) Number of People 18 425 19 370 20 345 21 295 22 260 23 205 24 180 25 110
220) The frequency distribution indicates the ages of 156 members of a health club. Age (years) Number of Members 20-30 48 30-40 40 40-50 44 50-60 12 60-70 0 70-80 8 80-90 4
24
221) The frequency distribution indicates the number of fish caught by each fisherman in a group of 50 fishermen. Number of Number of Fish Caught People 1 16 2 12 3 10 4 2 5 6 6 4
Construct a stem and leaf display for given data. 222) The ages of the instructors at a local college are given below. 36 61 57 62
46 38 34 45
43 42 35 49
58 49 46 55
223) The instructor of an introductory mathematics class has recorded the number of students attending his class in the last 4 weeks. Below are his recordings. 30 26 24 26 33 27 26 35 28 22 33 27 19 34 17 34 30 19 18 34 224) The numbers below represent the commute times (in minutes) for a group of college students. 11 16 12 16 13 25 26 35 2 12 23 12 16 34 21 4 7 24 23 34
25
Use the circle graph to solve the problem. 225) A survey of the 7321 vehicles on the campus of State University yielded the following circle graph.
11%
14%
35%
6% 5% 29% Find the number of motorcycles. Round your result to the nearest whole number. 226) The circle graph below gives the number of residents in the residence halls at the state university.
200 135
295
115
140 245 Write the ratio as a fraction in lowest terms of the number of residents at Adams to the number of students at Dodge.
26
227) A survey of the 8234 vehicles on the campus of State University yielded the following circle graph.
11%
14%
35%
9% 7% 24% What percent of the vehicles are hatchbacks? 228) There are 13,000 students attending the local university. The circle graph shows the percentage of those students who attend different sporting events.
4% 14%
40% 17%
25% What percentage of students do not attend Soccer or Volleyball matches?
27
Provide an appropriate response. 229)
How many people were 21 years old or older? 230)
How many games were attended by the students in the survey?
28
231) What is the modal class? Provide an appropriate answer. 232) Use the following frequency distribution to determine the modal class (or classes). Class Frequency 0-3 8 4-7 5 8-11 4 12-15 6 16-19 8 20-23 8 233) Use the following frequency distribution to determine the modal class (or classes). Class Frequency 15-19 5 20-24 3 25-29 5 30-34 7 35-39 6 40-44 7 234) Use the following frequency distribution to determine the midpoint of the second class. Class Frequency 15-17 6 18-20 4 21-23 6 24-26 8 27-29 7 30-32 5 Find the mean of the set of data. Round your answer to the nearest tenth. 235) 101, 34, 70, 70, 34, 101, 19, 19, 19, 101, 19, 19, 70
29
236) 6, 16, 13, 18, 12, 13, 1, 2, 10 237) 116, 48, 11, 60, 90, 25, 58, 84, 4 Find the median of the set of data. 238) 77, 28, 219, 149, 288, 234, 230 239) 11, 28, 38, 57, 60, 62, 80 240) 4, 7, 19, 24, 30, 39, 49 Find the midrange of the set of data. 241) 14, 20, 2, 4, 19, 20, 22, 4, 12 242) 11, 3, 10, 11, 11, 18, 10, 18, 18, 3, 10, 3, 18 243) 227, 221, 211, 211, 211, 211, 221, 216, 216, 216, 221, 221 Find the mode or modes for the set of numbers. 244) 113, 141, 156, 113, 188, 199, 162 245) 95, 25, 95, 13, 25, 29, 56, 95 246) 20, 34, 46, 34, 49, 34, 49 Solve the problem. 247) The table below gives the total spectator attendance for various sports. Sport Attendance (millions) Baseball 64.9 Basketball 27.7 Tennis 6.7 Golf 21.7 Football 36.9 Soccer 14.8 Hockey 17.1 Determine to the nearest tenth the midrange of these attendance numbers. 248) The weight in pounds of ten randomly-selected football players are as follows. Find the midrange of these weights. 245 304 310 251 195 185 230 264 315 196
30
249) State College's women's basketball team had the following scores in their last 7 games: 85, 66, 74, 98, 54, 85, and 83. Find the mean of these scores. Find the range for the set of data given. 250) 0.124 0.117 0.56 0.404 0.62 0.332 251) 72 148 35 104 174 Find the standard deviation. Round to one more place than the data. 252) 5, 17, 7, 6, 14, 8, 13, 18, 11, 24 253) 2, 3, 14, 20, 18, 20, 19, 8, 12 A company installs 5000 light bulbs, each with an average life of 500 hours, standard deviation of 100 hours, and distribution approximated by a normal curve. Find the percentage of bulbs that can be expected to last the period of time. Round to the nearest hundredth, if necessary. 254) Between 290 hours and 540 hours 255) Less than 520 hours Solve the problem. Round to the nearest hundredth, if necessary. 256) The average weekly income of teachers in one school district is $750 with a standard deviation of $85. Find the percentage of teachers earning more than $825 a week? 257) The average middle-distance runner at a local high school runs the mile in 5.6 minutes, with a standard deviation of 0.2 minute. Find the percentage of a runners that will run the mile in less than 5.2 minutes? 258) The monthly incomes of trainees at a local mill are normally distributed with a mean of $1200 and a standard deviation $160. What percentage of trainees earn less than $850 a month? 259) The volumes of soda in quart soda bottles are normally distributed with a mean of 32.3 oz and a standard deviation of 1.2 oz. Find the percentage of bottles with a volume of less than 32 oz? 260) Assume that the weights of quarters are normally distributed with a mean of 5.67 g and a standard deviation 0.070 g. A vending machine will only accept coins weighing between 5.45 g and 5.84 g. What percentage of legal quarters will be rejected? Solve the problem. 261) Which of the following are measures of dispersion: class mark, correlation coefficient, critical value, mean, median, midpoint, midrange, mode, range, standard deviation, z-score. 262) The mean and median salaries for U.S. workers are $28,000 and $43,000, respectively. State whether you think the distribution of salaries is normal, rectangular, skewed left, or skewed right. Explain your answer. 263) Do "average" and "mean" mean the same thing? Explain.
31
264) True or false - if false, explain why or modify the statement so that it becomes true: As the standard deviation of a data set increases, the width of the standard normal curve for the data set increases. Determine if a correlation exists at the indicated level of significance. 265) x y 14 28 17 40 α = 0.01 19 46 20 49
x 14 266) 16 17 19
y 63 43 47 27
α = 0.05
x 26 30 37 42 43
y 28 34 36 48 42
α = 0.05
x 16 17 21 24 25
y 10 22 28 37 43
α = 0.01
x 47 49 52 53 54
y 28 37 34 40 49
α = 0.05
267)
268)
269)
Assume that a sample of bivariate data yields the correlation coefficient, r, indicated. Use the critical values table for the specified sample size and level of significance to determine whether a linear correlation exists. 270) r = 0.285 when n = 82 at α = 0.01
32
271) r = 0.722 when n = 18 at α = 0.05 272) r = 0.223 when n = 5 at α = 0.05 Find the equation of the line of best fit from the data in the table. Round the slope and the y-intercept to the nearest hundredth. 273) x 12 14 16 18 20 y 54 53 55 54 56 274) x 24 y 15
26 13
28 20
30 16
32 24
1 3 5 7 9 275) x y 143 116 100 98 90 276) x y
2 15
4 37
6 60
8 75
10 94
277) x 1 2 3 4 5 y 17 20 19 22 21
6 24
278) x 3 5 7 15 16 y 8 11 7 14 20 279) x 10 20 30 40 50 y 3.9 4.6 5.4 6.9 8.3 Solve the problem. 280) The following table shows the team payroll (in millions of dollars) of several major league baseball teams along with their winning percentages for the 1999 season. Team Payroll (millions of $) 18.9 30.3 33.2 62.5 66.1 73.6 79.3 Winning percentage 0.395 0.475 0.589 0.595 0.617 0.636 0.475 Find the equation of the line of best fit for the team payroll and the winning percentage. 281) The following table shows the price of several models of new cars along with the average number of repairs each model requires over a 5-year period. Price of car when new (in $1000's) 11.3 12.6 17.5 20.9 26.2 Number of repairs in first 5 years 7.3 6.0 2.5 2.2 1.9 Find the equation of the line of best fit for the price and the number of repairs.
33
282) The following table shows the average number of practice hours per week, for a sample of professional golfers, along with their average tournament scores per round. Ave. practice hours per week 15.6 Ave. score per round 71.5
17.2 71.3
21.4 70.3
22.3 68.8
26.5 69.1
Find the equation of the line of best fit for the hours practiced per week and the average score per round. 283) The following table shows the March electric bill of selected homes in Atlanta along with the number of rooms in each home. Number of rooms 4 March electric bill $65
5 $78
5 $88
6 $85
7 $92
7 $102
8 $108
Determine whether there is a correlation between the number of rooms and the electric bills at α = 0.05. 284) The following table shows the average number of practice hours per week, for a sample of professional golfers, along with their average tournament scores per round. Ave. practice hours per week 16.3 Ave. score per round 70.5
18.4 71.4
20.2 70.1
22.6 69.4
24.5 68.4
Determine the correlation coefficient for the hours practiced per week and the average score per round. 285) The following table shows the price of several models of new cars along with the average number of repairs each model requires over a 5-year period. Price of car when new (in $1000's) 11.9 Number of repairs in first 5 years 7.2
13.9
19.5
22.4
28.1
5.9
2.3
2.4
1.7
Determine the correlation coefficient for the price and the number of repairs. Tell whether the statement is true or false. If false, give the reason. 286) {x|x is a counting number greater than 35} = {35, 36, 37, . . .} 287) {4, 14, 23, 7, 35} = {35, 14, 7, 32, 4} 288) 17 ∉ {16, 14, 13, . . ., 1} 289) 13 ∉ {x|x is an even counting number} 290) 5 ∈ {10, 15, 20, 25, 30} 291) {6} = {x|x is an even counting number between 8 and 14}
34
Determine whether the sets are equal, equivalent, both, or neither. 292) {brake} and {break} 293) {63, 18, 100} and {18, 100, 63} 294) {1/10, 2/10, 3/10} and {0.1, 0.2, 0.3} 295) {x | x is a whole number} and {x | x is an integer} 296) {L, M, N, O} and {l, m, n, o} 297) {4, 13} and {41, 3} Find n(A) for the set. 298) A = {6, 8, 10, 12, 14} 299) A = {100, 101, 102, . . ., 1000} 300) A = {x∣x ∈ N and 15 ≤ x ≤ 23} Express the set in roster form. 301) The set of all whole numbers greater than 6 and less than 10 302) The set of the days of the week 303) {x|x is a natural number multiple of 5} Identify the set as finite or infinite. 304) {2, 4, 6, 8, . . .} 305) {x|x is a fraction between 84 and 85} 306) The set of odd whole numbers less than 100
35
Solve the problem. 307) Use the following graph which shows the number of customer service calls to a major appliance manufacturer, in millions, for the years 2006-2012. Use the graph to represent the set in roster form.
2006
2007
2008
2009 2010 2011 2012 Year The set of years in which the number of customer service calls exceeded 1 million. 308) Use the following table, which shows the average price for a new beverage that is served at a coffee chain. Let the 10 selected regions represent the universal set. Use the list to represent the set in roster form. Region A B C D E F G H K L
Price $7.45 $6.78 $6.49 $5.62 $4.40 $3.66 $3.16 $2.77 $2.24 $1.95
{x|x is a region in which the average price of the new beverage is at most $6.00} 309) Use the following table, which shows the average price for a new beverage that is served at a coffee chain. Let the 10 selected regions represent the universal set. Use the list to represent the set in roster form. Region A B C D E F G H K L
Price $7.19 $6.84 $6.11 $5.87 $4.41 $3.91 $3.08 $2.87 $2.27 $1.92
The set of regions in which the average price for the new beverage is less than $3.50. 36
Write the set in set-builder notation. 310) {2, 4, 6, 8} 311) {24, 30, 36, 42,..., 84} 312) {15, 16, 17, 18} Let A = {1, 3, 5, 7} B = {5, 6, 7, 8} C = {5, 8} D = {2, 5, 8} U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the statement is true or false. 313) A ≠ {7, 5, 3, 1} 314) C ⊆ D 315) C ⊂ D If the statement is true for all sets C and D, write "true." If it is not true for all sets C and D, write "false." Assume that C ≠ ∅, U ≠ ∅, and C ⊂ U. 316) U ⊂ ∅ 317) ∅ ⊆ ∅ 318) ∅ ⊆ C Use ⊆, ⊈, ⊂, or both ⊂ and ⊆ to make a true statement. 319) {All states west of the Rocky Mountains} 320) {0} 321) {s, r, t}
{All states west of the Atlantic Ocean}
∅ {s, r, t}
List all subsets or determine the number of subsets as requested. 322) List all the subsets of {6}. 323) List all the subsets of {wolf, cat, sheep}. 324) Determine the number of subsets of {10, 11, 12} Let U = {all soda pops}, A = {all diet soda pops}, B = {all cola soda pops}, C = {all soda pops in cans}, and D = {all caffeine-free soda pops}. Describe the set in words. 325) A ∩ B ∩ D
37
326) A ∩ B 327) A' ∩ C Use the Venn diagram to list the set of elements in roster form. 328) Find A' ∪ B. P >
&
$
X
@
F
329) Find A ∪ B.
i
b
e
m
g
t
k
330) Find (A ∪ B)'.
7
x
q
5 1
f
Provide an appropriate response. 331) Let U represent the set of all national parks in the United States. Let A represent the set of national parks in Colorado. Describe A'. 332) Let U represent the set of prisoners in United States prisons. Let A represent the set of prisoners inTexas state prisons. Describe A'. 333) At Wilson High School 25 girls participate in soccer, 33 girls participate in basketball, and 15 girls participate in both soccer and basketball. How many girls participate in either soccer or basketball? Let U = {q, r, s, t, u, v, w, x, y, z} A = {q, s, u, w, y} B = {q, s, y, z} C = {v, w, x, y, z}. Determine the following. 334) A ∩ B'
38
335) A' ∪ B 336) C' ∩ A' 337) A ∪ (B ∩ C) Let U = {q, r, s, t, u, v, w, x, y, z} A = {q, s, u, w, y} B = {q, s, y, z} C = {v, w, x, y, z} Determine the following. 338) (A ∪ B')' ∩ C' 339) (B' ∩ C)' ∪ A 340) A ∩ (B ∪ C) Construct a Venn diagram illustrating the following sets. 341) Consider the following chart which shows teams that won at least 6 medals in wine-tasting competitions. Let the vineyards shown represent the universal set. Gold Silver Bronze Total Franklin (F) 14 12 8 34 Upper (U) 12 5 14 33 Springton (S) 8 4 12 24 Inland (I) 8 5 1 14 Parkway (P) 1 8 4 13 Greenville (G) 4 1 1 6 Let A = set of teams that won at least 33 medals. Let B = set of teams that won at least 8 gold medals. Let C = set of teams that won at least 5 silver medals.
39
342) Let U = {cheese (c), sausage (s), pepperoni (p), onion (o), garlic (g), mushroom (m)}. Let A be the set of the four most popular pizzas ordered at Village Pizza in March-April. Let B be the four most popular pizzas in February-March, and let C be the four most popular pizzas in January-February. Then A = {cheese (c), onion (o), garlic (g), mushroom (m)} B = {cheese (c), onion (o), sausage (s), mushroom (m)} C = {cheese (c), sausage (s), pepperoni (p), mushroom (m)}
Use set statements to write a description of the shaded area. Use union, intersection and complement as necessary. 343)
344)
345)
346)
40
Use Venn diagrams to determine whether the following statements are equal for all sets A and B. 347) A ∪ (B ∩ C)', A ∪ (B' ∪ C') 348) A ∩ (B ∪ C), (A ∩ B) ∪ C 349) A ∪ (B ∩ C), (B ∩ C) ∪ A 350) (A' ∪ B)', A ∩ B' Determine which region, I through VII, the indicated element belongs. 351) Determine in which region of the Venn diagram the letter in question would be placed.
c
41
352) During a special club meeting of the Garden Club, three items were voted on. The votes of nine members are shown in the table that follows. Determine in which region of the Venn diagram the member in question would be placed. The set labeled "Vote 1" represents the set of members who voted "yes" on vote 1, and so on. Member Vote 1 Vote 2 Vote 3 1. Marcus yes yes no 2. Patterson no no yes 3. Klein yes no yes 4. Myers no no yes 5. Parker no no yes 6. Patel yes yes yes 7. Smith yes yes no 8. Szabo yes no yes 9. Ruiz yes yes no
Myers 353) Determine in which region of the Venn diagram the letter in question would be placed.
J
42
354) During a special club meeting of the Garden Club, three items were voted on. The votes of nine members are shown in the table that follows. Determine in which region of the Venn diagram the member in question would be placed. The set labeled "Vote 1" represents the set of members who voted "yes" on vote 1, and so on. Member 1. Marcus 2. Patterson 3. Klein 4. Myers 5. Parker 6. Patel 7. Smith 8. Szabo 9. Ruiz
Vote 1 yes no yes no no yes yes yes yes
Vote 2 yes no no no no yes yes no yes
Vote 3 no yes yes yes yes yes no yes no
Marcus Indicate whether the statement is a simple or a compound statement. If it is a compound statement, indicate whether it is a negation, conjunction, disjunction, conditional, or biconditional by using both the word and its appropriate symbol. 355) The clown is not amusing. 356) It is false that whales are fish and bats are birds. 357) Trevor wanted to attend the meeting, but he had to go to the party. 358) The animal is a mammal if and only if it nurses its young. 359) The car is in the garage and the bicycle is in the driveway. Convert the compound statement into words. 360) p = "The food tastes delicious." q = "We eat a lot." r = "Nobody has dessert." (q ∨ p) ∧ r
43
361) p = "The food tastes delicious." q = "We eat a lot." r = "Nobody has dessert." ~q ∨ (p ∧ r) 362) p = "People buy large houses." q = "Fossil fuels pollute the air." q∧ p Write the compound statement in words. 363) Let p = "The monitor is included." q = "The color printer is optional." r = "The zip drive is extra." ~(p ∧ q) → (r ∨ ~q) 364) Let r = "The puppy is trained." p = "The puppy behaves well." q = "His owners are happy." (~r ∨ ~p) → ~q 365) Let r = "The puppy is trained." p = "The puppy behaves well." q = "His owners are happy." r ∧ (p → q) Write a negation of the statement. 366) Some athletes are musicians. 367) Some citizens obey traffic laws. 368) She earns more than me. Let p represent the statement, "Jim plays football", and let q represent "Michael plays basketball". Convert the compound statements into symbols. 369) Jim does not play football or Michael plays basketball. 370) It is not the case that Jim does not play football and Michael does not play basketball. 371) Jim plays football and Michael plays basketball. Write the compound statement in symbols. Let r = "The food is good," p = "I eat too much," q = "I'll exercise." 372) If the food is good or if I eat too much, I'll exercise. 373) The food is good and if I eat too much, then I'll exercise.
44
374) If I exercise, then I won't eat too much. 375) If the food is good, then I eat too much. Select letters to represent the simple statements and write each statement symbolically by using parentheses then indicate whether the statement is a negation, conjunction, disjunction, conditional, or biconditional. 376) Pat Rabbit did not go into Mr. McNoodle's garden and Molly collected blueberries. 377) The lights are on if and only if it is not midnight or it is wintertime. 378) If people drive small cars then people will use less fuel and the ozone hole will not expand. Construct a truth table for the statement. 379) ~(s ∨ t) ∧ ~(t ∧ s) 380) (s ∧ t) ∨ (~s ∧ ~t) 381) ~(~(q ∨ s)) Determine the truth value for the simple statement. Then use these truth values to determine the truth value of the compound statement. Use the chart or graph when provided. 382) 0 > -1 and 9 ≤ 10 383) 9 + 6 = 20 - 5 or 60 ÷ 5 = 3 · 4 Translate the statement into symbols then construct a truth table. 384) p = Parker will work in an office. q = Parker will work as a forest ranger. r = Parker will work as a landscape architect. Parker will not work in an office, but he will work as a forest ranger or a landscape architect. 385) p = The doctor prescribed medicine. q = The patient has recovered. The doctor did not prescribe medicine but the patient recovered. Construct a truth table for the statement. 386) ~(p ∧ q) → (p → (~s ∧ q)) 387) (~p ∨ ~q) → ~(q ∧ p) Determine the truth value for each simple statement. Then, using the truth values, give the truth value of the compound statement. 388) 7 < 10 or 8 > 4, and 9 < 10 389) 3 + 1 = 10 if and only if 9 = 14.
45
Given p is true, q is true, and r is false, find the truth value of the statement. 390) (~p ∧ q) ↔ ~r 391) (q ∨ r) ↔ (p ∧ q) Determine whether the statement is a self-contradiction, an implication, a tautology (that is not also an implication), or none of these. 392) (q ∧ p) ↔ ~(p ∧ q) 393) (p ∨ q) ∧ ~q Use the information given in the chart or graph to determine the truth values of the simple statements. Then determine the truth value of the compound statement given. 394)
The most common type of TV show watched by 13-29 year-olds is the sitcom, or 31% of 13-29 year-olds watch sports and 20% of 30-49 year-olds watch sports. 395) Planet X
● Moon 1 o o
Moon 2 Moon 3
Diameter of moons: o 4-7 km
○ Moon 4
○ 8-16 km
○ Moon 5
● 17-25 km
May have: o water ice
● atmosphere ○ both
If Moon 4 has a diameter of 8-16 km or Planet X has four moons, then Moon 1 may have water ice.
46
396) Planet X
● Moon 1 o o
Moon 2 Moon 3
○ Moon 4 ○ Moon 5
Diameter of moons: o 3-9 km
○ 10-18 km
● 19-24 km
May have: o water ice
● atmosphere ○ both
Moon 1 has a smaller diameter than Moon 3 and Moon 5 may have water ice, if and only if Moon 2 may have both water ice and an atmosphere. Write the compound statement in symbols. Then construct a truth table for the symbolic statement. Let r = "The food is good," p = "I eat too much," q = "I'll exercise." 397) The food is good and if I eat too much, then I'll exercise. 398) If the food is good, then I eat too much. 399) If I exercise, then I won't eat too much. Use DeMorgan's laws or a truth table to determine whether the two statements are equivalent. 400) (q → p) ∧ (p → q), (q ↔ p) 401) (p → q) ∨ (q → p), (p ↔ q) 402) ~(~p → q), p ∨ ~q Determine which, if any, of the three statements are equivalent. 403) I) If the garden needs watering, then the garden needs weeding or the garden is not lovely. II) The garden needs watering, and it is false that the garden does not need weeding and the garden is not lovely. III) The garden needs watering, and the garden needs weeding or the garden is lovely. 404) I) Jan is well or Jan is still recovering. II) If Jan is still recovering, then Jan is not well. III) If Jan is well, then Jan is not still recovering. 405) I) If it is sunny and the pool is open, then I will go swimming. II) If I do not go swimming, then it it is not the case that it is sunny or the pool is open. III) It is sunny and the pool is open, or I will go swimming. Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false. 406) If a line does not intersect a circle in more than one point; then the line is tangent to the circle.
47
407) If 10 does not divide the counting number, then 5 does not divide the counting number.
408) If
1 is not an integer, then n is not an integer. n
Write an equivalent sentence for the statement. 409) It is not true that you are a day late and a dollar short. (Hint: Use De Morgan's laws.) 410) If you are wealthy then you are wise, and if you are wise then you are wealthy. (Hint: Use the fact that (p → q) ∧ (q → p) is equivalent to p ↔ q.) 411) If it is raining, you take your coat. (Hint: Use the fact that p → q is equivalent to ~p ∨ q.) Write the indicated statement. Use De Morgan’s Laws if necessary. 412) If it is a cat, then it catches birds. Inverse 413) If I pass, I'll party. Contrapositive 414) If x = 4, then x2 = 16. Converse Determine if the argument is valid or invalid. Give a reason to justify answer. 415) If it rains, then the squirrels hide. The squirrels are hiding. ∴ It is raining. 416) If the bell rings, then we answer the door. The bell rings. ∴ We answer the door. 417) If the bough breaks, then the cradle will fall. The bough breaks. ∴ The cradle will fall. Use the method of writing each premise in symbols in order to arrive at a valid conclusion. 418) All fish can dream. Any dead animal is unable to dream. All live animals have a heartbeat. Therefore, ... 419) Hard workers sweat. Sweat brings on a chill. Anyone who doesn't have a cold never felt a chill. Anyone who works doesn't have a cold. Therefore, ...
48
Use an Euler diagram to determine whether the syllogism is valid or invalid. 420) Not all cars are considered sporty. Not all cars are safe at high speeds. ∴ Sports cars are safe at high speeds. 421) All painters use paint. All painters use brushes. Some people who use paint are teachers. ∴ Some painters are teachers. Construct a truth table to determine when the lightbulb is on. That is, determine which switches must be open and which switches must be closed for the lightbulb to be on. 422)
Represent each circuit with a symbolic statement. Using a truth table, state whether the circuits are equivalent. 423)
,
49
Answer Key Testname: 1333-PT-FINAL
1) 8 ways 34 ways 2) 6 teams: 5 + 4 + 3 + 2 + 1 = 15 games 7 teams: 6 + 5 + 4+3+2+1+ = 21 games n teams : n(n - 1) 2 games 16 teams: 120 games 3)
24) $104.00 25) 31 26 33 32 30 28 27 34 29 26) 27 22 23 20 24 28 25 26 21 27) 0 57 28) 80 29)
4) 30) 31) 5)
32) 33) 34)
6) 9999 × 9999 = 99,980,001 7) 40,000 - 6789 = 33,211 8) -4 9) 24 10) W 11) t4 = 10 12) B 13) $20.40 14) C 15) 5 gal 16) 20 teaspoons 17) $61.73 18) $14.50 19) 1400 ft, 2800 ft 20) 14 drinks 21) 7 hr 22) 73 newspapers 23) $85
35) 36)
11 36 2 5 2 5 1 4165 6 4165 5 1274 1277 649740 1 649740
37) 0.1212 38) 0.0364 39) 0.5758 40) 0.0364 41) 0.1515 42) 0.4909 43) 0 44) 0.2182 45) 0.42 46) 0.4758 47) 0.3125 48) 0.0080 49) 0.0536 50) 0.9913 51) 0.433 52) 0.3070
53) 0.0185 54) 0.9815 55) 0.1990 56) 0.275 57) 0.130 58) 0.230 31 59) 365 60)
1 3
61)
9 22
62)
1 2
63)
1 4
64)
1 6
65)
7 15
66)
8 9
67)
4 13
68)
2 3
69)
3 7
70)
5 18
71)
9 13
72)
5 18
73)
334 365
74) 0.768 75) 0.3777 10 76) 13 77) 1:5 78) 1:4 79) 1:1 80) 1:7 81) 3:5 82) 2:3
83)
6 7
84)
4 9
85)
7 9
86)
5 11
87) 1 to 3 88) 99 to 1 89) 3 : 6 90) 4 to 1 91) $23,100 92) 0.86 93) $31,000 at age 70 94) 60: $8400 70: $19,200 80: $22,000 95) 3 96) $23,705 97) 2 98) $300 99) 1.5 100) 0.50 101) $8000.00 102) $61.62 103) -$0.23 104) $0.00 105) $0.00 106) -$0.50 107) -$5.00 108) -$0.50 109) $6.50 110) -$1 111) 12 112) 11 113) 57 114) 64 115) 24 116) 24 117) 24 1 118) 3 1 119) 3 120)
50
2 9
121)
1 3
122) 0 5 123) 18 124)
1 16
125) hhtt, htht, htth, thht, thth, tthh 126) BBBB, BBBG, BBGB, BBGG, BGBB, BGBG, BGGB, BGGG, GBBB, GBBG, GBGB, GBGG, GGBB, GGBG, GGGB, GGGG 127) 3 H, 3 T, 4 H, 4 T, 9 H, 9 T 128) (3, 1),(3, 2),(3, 4),(3, 5),(3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3) 129) (3, 1),(3, 2),(3, 3), (3, 4),(3, 5),(3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3) 130) KQ, KA, Q K, Q A, A K, A Q 131) 1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 2 1, 2 2, 2 3, 2 4, 2 5, 2 6, 5 1, 5 2, 5 3, 5 4, 5 5, 5 6 7 132) 10 133)
4 663
134) 4.5% 1 135) 16 136) 2.8% 1 137) 125 138)
4 7
Answer Key Testname: 1333-PT-FINAL
139)
3 10
164)
43 87
140)
5 6
165)
43 48
141)
31 52
166)
2 5
142)
1 2
143) 0.554 144) 0.84 145) 1 10 146) 13 147) P(A or B) = 0.3 148) P(A) = 0.6 11 149) 51 150) 0 1 151) 6 152)
1 3
153)
11 50
4 154) 17 155)
1 7
156)
1 7
157)
3 28
158)
3 7
1 159) 28 160)
1 7
161)
5 11
162)
76 87
163)
6 17
167) 168) 169) 170) 171) 172) 173) 174) 175) 176) 177) 178) 179) 180) 181) 182) 183) 184) 185) 186) 187) 188) 189) 190) 191) 192) 193) 194) 195)
120 144 5,040 6 120 40 30 60 45,360 6,760,000 54,432,000 840 210 30,240 1 67,600 1000 2,997 33,800 210 56 120 8 980 60 0 57 931 31 56 196) 11 197)
3 2
198) 70
199) Answers will vary. One possibility is: I would not be comfortable claiming a Democrat victory because so many people in the survey are undecided. 200) Descriptive statistics. 201) Her poll may be biased. In particular, it is probably biased towards that portion of the population that is old. To improve the poll, she must poll persons in other age ranges and in different circumstances of employment and education. 202) Convenience sampling 203) Stratified sampling 204) Systematic sampling 205) There are many more adult drivers than teenage drivers, so it is reasonable that more adults have accidents.
51
206) Average values give no information at all about the range of values in the population. 207) There are many more swimmers at ocean beaches than at lake beaches, so it is reasonable that there are more drownings at the ocean beaches. 208) Answers may vary. One possible solution:
209) Answers may vary. One possible solution:
200 8 200 9 201 0 201 1 201 2 Year
199 0 199 5 200 0 200 5 201 0 Year
Answer Key Testname: 1333-PT-FINAL
210) Answers may vary. One possible solution:
214)
218) Frequency
220)
224)
Number of Novels Frequency 0 1 2 3 4 5 Weight in 6 ounces 7 219) 215) Frequency
48
24
227) 228) 229) 230) 231) 232)
500 200 8 200 250
9 201 0 201 1
Weight in pounds 216) Frequency 221)
201 2
16
Year 211) Class Frequency 34-36 9 37-39 6 40-42 5 212)
8
Hours of bulb life 217) Frequency
Class Frequency 43-44 3 45-46 1 47-48 3 49-50 6 51-52 2 213)
222) 3 4 5 6
Height in Number of Accidents Frequency 0 1 2 3 4 5
inches
4568 2356699 578 12
223) 1 2 3 7899 24666778 00334445
52
0 1 2 3 247 12223666 133456 445 225) 805 23 226) 59
233) 234) 235) 236) 237) 238) 239) 240) 241) 242) 243) 244) 245) 246) 247) 248) 249) 250) 251) 252) 253) 254) 255) 256) 257) 258) 259) 260) 261)
35% 58% 20 people 72 18 years 0-3; 16-19; 2023 30-34; 40-44 19.0 52.0 10.1 55.1 219 57 24 12.0 10.5 219.0 113 95 34 35.8 million 250 lb 77.9 0.503 139 6.1 7.1 63.75% 57.93% 18.94% 2.28% 1.43% 40.13% 0.86% Range and standard deviation.
Answer Key Testname: 1333-PT-FINAL
262) Since the median exceeds the mean, the distribution is skewed left. 263) No. "Average" can denote any of the following: mean, median, mode, or midrange. 264) False. The independent variable in the standard normal curve , z, measures how far a data point lies from the mean in units of numbers of standard deviations. So, the standard normal distribution is always the same width regardless of the value of the standard deviation of the data. 265) Correlation exists. 266) Correlation exists. 267) Correlation exists. 268) Correlation exists. 269) Correlation does not exist. 270) Yes 271) Yes 272) No 273) y = 0.25x + 50.4
274) y = 1.05x - 11.8 275) y = -6.2x + 140.4 276) y = 9.8x - 2.6 277) y = 1.17x + 16.4 278) y = 0.75x + 5.07 279) y = 0.11x + 2.49 280) y = 0.00211x + 0.431 281) y = -0.365x + 10.4 282) y = -0.254x + 75.4 283) Yes 284) -0.8633 285) -0.9195 286) False; 35 is not greater than 35. 287) False; the elements are not the same. 288) True 289) True 290) False; 5 is not an element of the set. 291) False; 6 is less than 8 292) Equivalent 293) Both 294) Both 295) Equivalent 296) Equivalent 297) Equivalent 298) n(A) = 5 299) n(A) = 901 300) n(A) = 9 301) {7, 8, 9} 302) {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} 303) {5, 10, 15, . . .} 304) Infinite 305) Infinite 306) Finite
307) {2006, 2007, 2008, 2010, 2011} 308) {D, E, F, G, H, K, L} 309) {G, H, K, L} 310) {x | x is an even natural number less than 10} 311) {x | x is a multiple of 6 between 18 and 90} 312) {x | x is an integer between 14 and 19} 313) False; the elements in A are the same as those listed. 314) True 315) True 316) False 317) True 318) True 319) ⊂ and ⊆ 320) ⊈ 321) ⊆ 322) {6}, { } 323) {wolf, cat, sheep}, {wolf, cat}, {wolf, sheep}, {cat, sheep}, {wolf}, {cat}, {sheep}, { } 324) 8 325) All diet, caffeine-free cola soda pops 326) All diet cola soda pops 327) All non-diet soda pops in cans 328) {X, P, @, F, &} 329) {e, b, g, i, m, t} 330) ∅ 53
331) The set of national parks that are not in the state of Colorado. 332) The set of United States prisoners that are not in Texas state prisons. 333) 43 334) {u, w} 335) {q, r, s, t, v, x, y, z} 336) {r, t} 337) {q, s, u, w, y, z} 338) ∅ 339) {q, r, s, t, u, w, y, z} 340) {q, s, w, y} 341)
346) 347) 348) 349) 350) 351) 352) 353) 354) 355)
356) 357) 358)
359) 360)
361)
362) 342)
363)
343) B ∩ A' 344) (A ∪ B) ∪ C' 345) A' ∩ C' ∩ B
A' ∪ B equal not equal equal equal I VII II II Simple statement, negation; ~ Compound; negation; ~ Compound; conjunction; ∧ Compound; biconditional; ↔ Compound; conjunction; ∧ We eat a lot or the food tastes delicious and nobody has dessert. We do not eat a lot or the food tastes delicious and nobody has dessert. Fossil fuels pollute the air and people buy large houses. It is not true that if the monitor is included and the color printer is optional, then the zip drive is extra or the color printer is not optional.
Answer Key Testname: 1333-PT-FINAL
364) If the puppy is not trained or the puppy does not behave well, then his owners are not happy. 365) The puppy is trained, and if the puppy behaves well then his owners are happy. 366) No athletes are musicians. 367) All citizens do not obey traffic laws. 368) She does not earn more than me. 369) ~p ∨ q 370) ~(~p ∧ ~q) 371) p ∧ q 372) (r ∨ p) → q 373) r ∧ (p → q) 374) q → ~p 375) r → p 376) ~p ∧ q; conjunction 377) p ↔ (~q ∨ r); biconditional 378) p → (q ∧ ~r); conditional 379) s t ~(s ∨ t) T
T
T
F
F F F
T F
F
F T
380) s
t
T
T
T
F
(s ∧ t) ∨ T F
F
T
F
F
381) q
s
F T ~(~(q ∨
T T T F F T F F 382) True 383) True 384) p q
r
T T T T F F F F 385) p
T F F F T T F F q
T F F F T F F F T T F T T T F F ~p ∧ q
T T
T F
F F
F F 386) p
T F q
s
T F ~(p
T T T T F F F F 387) p
T T F F T T F F q
T T F T T F F F T T F T T T F T (~p ∨ ~
T T T F F T F F 388) True 389) True 390) False
391) True 392) Self-contradict ion 393) None of these 394) True 395) False 396) True 397) r ∧ (p → q)
T T T F
T T T T
~p
r
p
q
T T T F T T T T F F F F F F F F
T
T
T
F
F
T
F
F
T
T
T
F
F
T
F
F
r
398) r → p r p T T F F
T F T F
r T F T T
399) q → ~p p q T T F F
T F T F
q F T T T
400) 401) 402) 403)
Equivalent Not equivalent Not equivalent II and III are equivalent 404) II and III are equivalent 54
405) None are equivalent 406) If a line is not tangent to a circle, then the line intersects the circle in more than one point. false 407) If 5 divides the counting number, then 10 divides the counting number. false 408) If n is an integer, then 1 is an n integer. false 409) You are not a day late or not a dollar short. 410) You are wealthy if and only if you are wise. 411) It is not raining or you take your coat. 412) If it's not a cat, it doesn't catch birds. 413) If I don't party, I didn't pass. 414) If x2 = 16, then x = 4. 415) Invalid by fallacy of the converse 416) Valid by the law of detachment 417) Valid by the law of detachment 418) All fish have a heartbeat.
419) Hard workers don't go to work. 420) Invalid 421) Invalid 422) The lightbulb will be on in all cases except when p and r are open and q is closed. p q r (~q ∨ r) T T T T T T F F T F T T T F F T F T T T F T F F F F T T F F F T 423) (p ∧ ~r) ∧ [(p ∧ q) ∨ r]; (p ∨ ~r) ∧ [(p ∧ q) ∨ r]; not equivalent
