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MATH 225N WEEK 4 PROBABILITY QUESTIONS AND ANSWERS 100% CORRECT

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2023/2024

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Download MATH 225N WEEK 4 PROBABILITY QUESTIONS AND ANSWERS 100% CORRECT and more Exams Mathematics in PDF only on Docsity! M ATH 225N WEEK 4 PROBABILITY QUESTIONS AND ANSWERS 100% CORRECT 1. Patricia will draw 8 cards from a standard 52-card deck with replacement. Which of the following are not events in this experiment? Select all that apply: drawing 8 hearts drawing 8 cards drawing 1 card drawing 3 kings (Events are any combinations of outcomes or particular results in an experiment. Drawing 8 cards is the experiment and drawing 1 card is a trial of the experiment, neither of which specify a result or outcome) 2. Which of the following gives the definition of event? Select the correct answer below: the set of all possible outcomes of an experiment a subset of the set of all outcomes of an experiment a planned activity carried out under controlled conditions one specific execution of an experiment 3. Which of the following gives the definition of trial? Select the correct answer below: a particular result of an experiment one specific execution of an experiment a planned activity carried out under controlled conditions the set of all possible outcomes of an experiment 4. Paul will roll two standard dice simultaneously. If Event A = both dice are odd and Event B = at least one die is even, which of the following best describes events A and B? Select two answers. Select all that apply: Mutually Exclusive Not Mutually Exclusive Independent Select the correct answer below: drawing a face card and then drawing a 3 without replacement from a standard deck of cards rolling a sum of 6 from the first two rolls of a standard die and a sum of 4 from the second two rolls drawing a 2 and drawing a 4 with replacement from a standard deck of cards drawing a heart and drawing another heart with replacement from a standard deck of cards 3. Which of the following shows mutually exclusive events? Select the correct answer below: rolling a sum of 9 from two rolls of a standard die and rolling 2 for the first roll drawing a red card and then drawing a black card with replacement from a standard deck of cards drawing a jack and then drawing a 7 without replacement from a standard deck of cards drawing a 7 and then drawing another 7 with replacement from a standard deck of cards 4. Which of the pairs of events below is mutually exclusive? Select the correct answer below: drawing an ace of spades and then drawing another ace of spades without replacement from a standard deck of cards drawing a 2 and drawing a 4 with replacement from a standard deck of cards drawing a heart and then drawing a spade without replacement from a standard deck of cards drawing a jack and then drawing a 7 without replacement from a standard deck of cards 4.1 Beth is performing an experiment to check if a die is fair. She rolls the die 5 times and records the sequence of numbers she gets. Which of these is an outcome of this experiment? Select all correct answers. Select all that apply: Rolling a die Rolling a die five times Rolling the sequence 1,1,2,1,6 Rolling five 4's Rolling the sequence 1,1,2 (An outcome is a specific result of an experiment. So the outcomes of this experiment are all the possible sequences of five die rolls. So in this case, a particular sequence such as 1,1,2,1,6 is an outcome. So is rolling five 4's, because this is a specific outcome (4,4,4,4,4) (Mutually exclusive events are events that cannot occur together. In this case, drawing an ace of spades and then drawing another ace of spades without replacement from a standard deck of cards are two events that cannot possibly occur together.) 5. A deck of cards contains RED cards numbered 1,2,3,4,5,6, BLUE cards numbered 1,2,3,4,5, and GREEN cards numbered 1,2,3,4. If a single card is picked at random, what is the probability that the card has an ODD number? Select the correct answer below: 10/15 8/15 14/15 6/15 (By counting, we can see that there are 8 odd cards, and a total of 15 cards in the deck. So the probability is 8/15). 6. Hector is a baseball fan but wants to watch something different. There are 5 basketball games, 2 football games, and 4hockey games that he can choose to watch. If Hector randomly chooses a game, what is the probability that it is a basketball game? • Give your answer as a fraction. Provide your answer below: 5/11 7. There are 26 cards in a hat, each of them containing a different letter of the alphabet. If one card is chosen at random, what is the probability that it is not between the letters L and P, inclusive? Provide your answer below: 21/26 8. A spinner contains the numbers 1 through 80. What is the probability that the spinner will land on a number that is not a multiple of 12? • Give your answer in fraction form. Provide your answer below: 74/80 9. An art collector wants to purchase a new piece of art. She is interested in 5 paintings, 6 vases, and 2 statues. If she chooses the piece at random, what is the probability that she selects a painting? Provide your answer below: 0.170 (This probability can be found using the binomial distribution with success probability p=0.55 and 14 trials. To find the probability that exactly 9 of the tosses are heads, use a calculator or computer: P(X=9)=binompdf(14,0.55,9)≈0.170. 15. Identify the parameter p in the following binomial distribution scenario. The probability of buying a movie ticket with a popcorn coupon is 0.546 and without a popcorn coupon is 0.454. If you buy 27 movie tickets, we want to know the probability that exactly 15 of the tickets have popcorn coupons. (Consider tickets with popcorn coupons as successes in the binomial distribution.) Select the correct answer below: 0.152 0.454 0.546 0.848 16. A softball pitcher has a 0.64 probability of throwing a strike for each pitch. If the softball pitcher throws 20 pitches, what is the probability that exactly 13 of them are strikes? • Round your answer to three decimal places. 0.184 This probability can be found using the binomial distribution with success probability p=0.64 and 20 trials. To find the probability that exactly 13 of the pitches are strikes, use a calculator or computer: P(X=13)=binompdf(20,0.64,13)=0.184. 17. Identify the parameter n in the following binomial distribution scenario. A basketball player has a 0.429 probability of making a free throw and a 0.571 probability of missing. If the player shoots 20 free throws, we want to know the probability that he makes no more than 12 of them. (Consider made free throws as successes in the binomial distribution.) Select the correct answer below: 8 © 12
© 20
© 32
18.Give the numerical value of the parameter p in the following binomial distribution scenario. A softball pitcher has a 0.675 probability of throwing a strike for each pitch and a 0.325 probability of throwing a ball. If the softball pitcher throws 29 pitches, we want to know the probability that exactly 19 of them are strikes. Consider strikes as successes in the binomial distribution. Do not include p= in your answer. Provide your answer below: 0.675 (The parameters p and n represent the probability of success on any given trial and the total number of trials, respectively. In this case, success is a strike, so p=0.675) 19. Identify the parameters p and n in the following binomial distribution scenario. Jack, a bowler, has a 0.38 probability of throwing a strike and a 0.62 probability of not throwing a strike. Jack bowls 20times (Consider that throwing a strike is a success.) Select the correct answer below: p=0.38,n=0.62 p=0.38,n=10 p=0.38,n=20 p=0.62,n=10 p=0.62,n=20 (In a binomial distribution, there are only two possible outcomes. p denotes the probability of the event or trial resulting in a success. In this scenario, it would be the probability of Jack bowling a strike, which is 0.38. The total number of repeated and identical events or trials is denoted by n. In this scenario, Jack bowls a total of 20 times, so n=20). 20.The Stomping Elephants volleyball team plays 30 matches in a week-long tournament. On average, they win 4 out of every 6matches. What is the mean for the number of matches that they win in the tournament? Select the correct answer below: Select the correct answer below: 5 14 19 24 23. Give the numerical value of the parameter n, the number of trials, in the following binomial distribution scenario. A weighted coin has a 0.486 probability of landing on heads and a 0.514 probability of landing on tails. If you toss the coin 27 times, we want to know the probability of getting heads exactly 11 times. Consider a toss of heads as success in the binomial distribution. Provide your answer below: 27 24.The probability of winning on an arcade game is 0.659. If you play the arcade game 30 times, what is the probability of winning exactly 21 times? • Round your answer to two decimal places. Provide your answer below: .14 25.The probability of buying a movie ticket with a popcorn coupon is 0.526. If you buy 26 movie tickets, what is the probability that exactly 15 of the tickets have popcorn coupons? • Round your answer to three decimal places. Provide your answer below: .137 26.The probability of buying a movie ticket with a popcorn coupon is 0.608. If you buy 10 movie tickets, what is the probability that more than 3 of the tickets have popcorn coupons? (Round your answer to 3 decimal places if necessary.) Participants stoppedsmoking did not stopsmoking Total given e-cigarette 11 21 32 not given e-cigarette 0 16 16 Total 11 37 48 Provide your answer below: .951 27. A softball pitcher has a 0.507 probability of throwing a strike for each pitch. If the softball pitcher throws 15 pitches, what is the probability that more than 8 of them are strikes? (Round your answer to 3 decimal places if necessary.) Provide your answer below: .323 28. A 2014 study by researchers at the University College Antwerp and the University of Leuven showed that e-cigarettes are effective at reducing cigarette craving. Participants were separated into two groups. One group was given e-cigarettes and the other was told to not smoke e-cigarettes. Two months later, researchers observed how many participants had stopped smoking cigarettes. The following table shows approximate numbers. According to the table, what is the probability that a randomly chosen participant did not stop smoking, given that the participant received an e-cigarette? Select the correct answer below: 21/48 11/48 11/32 16/16 0/16 21/32 29.Researchers wanted to study if having a long beak is related to flight in birds. They surveyed a total of 34 birds. The data are shown in the contingency table below. What is the relative risk of flying for those birds that have long beaks? Round your answer to two decimal places. • Marginal distributions are the row and column percentages. Breakfast and lunch are in the rows, so use the row totals to determine the percentages. The "Breakfast" percentage is 311/994≈0.31, and the "Lunch" percentage is 683/994≈0.69. The marginal distribution is 31%, 69%. 31. 155 fitness center members were asked if they run and if they lift weights. The results are shown in the table below. Does not Run Runs Total Does not Lift Weights 30 68 98 Lifts Weights 16 41 57 Total 46 109 155 Given that a randomly selected survey participant does not run, what is the probability that the participant lifts weights? • Enter the answer as a fraction. Provide your answer below: 16/46 = 8/23 32.Fill in the following contingency table and find the number of students who both have a cat AND have a dog. Provide your answer below: 35 33.Researchers wanted to study if having a long beak is related to flight in birds. They surveyed a total of 34 birds. The data are shown in the contingency table below. What is the odds ratio for birds that fly having long beaks against birds that do not fly having long beaks? Round your answer to two decimal places. Flies Does not fly Total Long beak 11 3 14 Students Have a dog Do not have a dog Total Have a cat 35 25 60 Do not have a cat 27 11 38 Total 62 36 98 Not a long beak 7 13 20 Total 18 16 34 Provide your answer below: 6.81 (The odds that a bird that flies also has long beak are 11 to 7. The odds that a bird that does not fly also has long beak are 3to 13. The odds ratio is then 11/17 / 3/13≈6.81. In this study, birds that fly had almost 7 times the odds of also having long beaks as the birds that do not fly.) 34.Fill in the following contingency table and find the number of students who both watch comedies AND watch dramas. Students Watch dramas Do not watch dramas Total Watch comedies 16 25 41 Do not watch comedies 38 27 65 Total 54 52 106 Provide your answer below: 16 35. Researchers wanted to study if couples having children are married. They surveyed a large group of people. The data are shown in the contingency table below. What is the odds ratio for married people having children against unmarried people having children? Round your answer to two decimal places. Children No Children Total Married 97 35 132 Not Married 68 71 139 Total 165 106 271 The odds that a married couple has children are 97 to 35. The odds that an unmarried couple has children are 68 to 71. The odds ratio is then 97/35 ÷ 68/71 ≈ 2.89. In this study, people who are married had about 3 times the odds of having children as people who are not married. 36. Doctors are testing a new antidepressant. A group of patients, all with similar characteristics, take part in the study. Some of the patients receive the new drug, while others receive the Afraid of heights Not afraid of heights Total Afraid of flying 76 33 109 Not afraid of flying 82 370 452 Total 158 404 561 Provide your answer below: 5.87 (The probability that someone with a fear of heights is afraid of flying is 76/158=38/79. The probability that someone who does not have a fear of heights is afraid of flying is 33/403. The relative risk is then 38/79 /v 33/403≈5.87. This means that in this survey, people with a fear of heights were 587% as likely to have a fear of flying as people without a fear of heights. 40. A study of drivers with speeding violations in the last year and drivers who use cell phones produced the following fictional data: Violatio n No violation Tota l Cell phone user 25 280 305 Not a cell phone user 45 405 450 Total 70 685 755 Find the probability that a driver received a violation, given that the driver is a cell phone user. Select the correct answer below: 280/305 70/755 25/305 25/70 41.Find the probability that a randomly chosen person takes public transit to work given that the person does not support the environmental bill. People Drive to work Walk to work Public Transport to work Tota l Support bill 5 30 20 55 Do not support bill 20 3 10 33 Total 25 33 30 88 Give your answer as a fraction. 10/33 42.Fill in the following contingency table and find the number of students who both do not go to the beach AND do not go to the mountains. Provide your answer below: 10 43.Fill in the following contingency table and find the number of students who both have a cat AND have a dog. Provide your answer below: 14 44.Researchers wanted to study if couples having children are married. They surveyed a large group of people. The data are shown in the contingency table below. What is the odds ratio for people having children to be married against people not having children to be married? Round your answer to two decimal places. Students Go to the mountai ns do not go to the mountains Tot a l Go to the beach 32 22 54 Do not go to the beach 17 10 27 Total 49 32 81 Students Have a dog Do not have a dog Total Have a cat 14 35 49 Do not have a cat 32 17 49 Total 46 52 98 flu flu l New Vaccine 15 375 390 Traditional Vaccine 55 225 280 Total 70 600 670 Answer: 6.11 The odds that a person who receives the new vaccine does not catch the flu is 375 to 15, or 25 to 1. The odds that a person who receives the traditional vaccine does not catch the flu is 225 to 55, or 45 to 11. The odds ratio is then 25/1 ÷ 45/11≈6.11. In this experiment, people who took the new vaccine had just over 6 times the odds of not catching the flu as people who did not take the new vaccine. 47.Doctors are testing a new antidepressant. A group of patients, all with similar characteristics, take part in the study. Some of the patients receive the new drug, while others receive the traditional drug. During the study, a number of patients complain about insomnia. The data are shown in the contingency table below. What is the relative risk of insomnia for those who receive the new drug? Round to two decimal places. Insomni a No insomnia Tota l New drug 52 226 278 Traditional drug 36 295 331 Total 88 521 609 Answer: 1.72 (The probability that a patient who receives the new drug develops insomnia is 52/278=26/139. The probability that a patient who receives the traditional drug develops insomnia is 36/331. The relative risk is then 26/139 ÷ 36/331≈1.72. This means that in this study, people who took the new drug were 172% more likely to develop insomnia. 48. In a recent survey, a group of people were asked if they were happy or unhappy with the state of the country. The data are shown in the contingency table below, organized by political party. What is the odds ratio for people unhappy with the state of the country to be republicans against people happy with the state of the country to be republicans? Round your answer to two decimal places. Unhappy Unhappy Total Republican 152 98 250 Democrat 104 146 250 Total 256 244 500 Provide your answer below: 0.46 (The odds that a person unhappy with the state of the country is a republican are 98 to 146, or 49 to 73. The odds that a person happy with the state of the country is a republican are 152 to 104, or 19 to 13. The odds ratio is then 49/73 ÷ 19/13≈0.46. In this study, a person unhappy with the state of the country had about half the odds of being a republican as a person happy with the state of the country. 49.Researchers wanted to study if couples having children are married. They surveyed a large group of people. The data are shown in the contingency table below. What is the relative risk of being married for those who have children? Round your answer to two decimal places. Children No Children Total Married 97 35 132 Not Married 68 71 139 Total 165 106 271 Answer: 1.78 (he probability that someone who has children is married is 97/165. The probability that someone who does not have children is married is 35/106. The relative risk is then 97/165 ÷ 35/106≈1.78. This means that in this survey, people who have children were 178%as likely to be married as people who do not have children. 50.Kelsey, a basketball player, hits 3-point shots on 38.1% of her attempts. If she takes 14 attempts at 3-point shots in a game, what is the probability that she hits exactly six of them? Use Excel to find the probability. • Round your answer to three decimal places. Provide your answer below: 0.198 51. A computer graphics card manufacturer is testing an improvement to its production process. If a sample of 100 graphics cards manufactured using the new process has a less than 10% chance of having 3 or more defective graphics cards, then the manufacturer will switch to the new process. Otherwise, the manufacturer will stay with its existing process. If the probability of a defective graphics card using the new process is 0.9%, will the manufacturer switch to the new production process? Select the correct answer below: Yes, because the probability of having 3 or more defective graphics cards is greater than 0.10. Yes, because the probability of having 3 or more defective graphics cards is less than 0.10. No, because the probability of having 3 or more defective graphics cards is less than 0.10. No, because the probability of having 3 or more defective graphics cards is greater than 0.10. (Note that this is a cumulative binomial probability. In this case, we want to find the probability of 3 or more successes, inclusive, where a success is one of the graphics cards being defective. The probability of having 2 or fewer defective graphics cards is the complement of the probability of having 3 or more defective graphics cards. To determine the probability from a binomial distribution using Excel, follow the steps below. 1. First press FORMULAS and then INSERT FUNCTION. 2.Then select the BINOM.DIST function. 3. Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 2, 100, and 0.009, in that order. defective? Use Excel to find the probability. • Round your answer to three decimal places. 0.970 56. A database system assigns a 32-character ID to each record, where each character is either a number from 0 to 9 or a letter from A to F. Assume that each number or letter being selected is equally likely. Find the probability that at least 20characters in the ID are numbers. Use Excel to find the probability. • Round your answer to three decimal places. 0.578 57. A fair spinner contains the numbers 1, 2, 3, 4, and 5. For an experiment, the spinner will be spun 5 times. If Event A = the spinner lands on numbers all less than 3, what is an outcome of Event A? a total sum less than 4 spinner lands on 1, 3, 1, 2, 1 a total sum of 11 spinner lands on 1, 2, 1, 2, 2 58. A poll is conducted to determine if political party has any association with whether a person is for or against a certain bill. In the poll, 214 out of 432 Democrats and 246 out of 421 Republicans are in favor of the bill. Assuming political party has no association, the probability of these results being by chance is calculated to be 0.01. Interpret the results of the calculation. We can expect that 246 out of every 421 Republicans are in support of this bill. We cannot say the results are statistically significant at the 0.05 level. At the 0.01 level of significance, political party is associated with whether a person supports this bill. At the 0.01 level of significance, political party determines whether a person supports this bill. (Statistical significance of any level does not mean that there is a certain factor that affects the results of the experiment) 59. Arianna will roll a standard die 10 times in which she will record the value of each roll. What is a trial of this experiment? one roll of the die rolling at least one 5 ten rolls of the die rolling a sum of 40 60. A health survey determined the mean weight of a sample of 762 men between the ages of 26 and 31 to be 173 pounds, while the mean weight of a sample of 1,561 men between the ages of 67 and 72 was 162 pounds. The difference between the mean picking 10 people out of a crowd and having them all have brown eyes guessing somebody's name correctly without having previously met the person 63.Before a college professor gave an exam, he held a review session, where 30 of his 150 students attended the review. The mean score of the students who attended was 86%, whereas the mean score of the students who didn’t attend the review was 79%. The difference in the mean scores is significant at the 0.05 level, assuming the review session does not associate with a higher exam score. Determine the meaning of this significance level. Select the correct answer below: It is not unusual to see the mean exam score of 120 students be 79% because the testing abilities of students vary. We expect the mean score of a group of 30 students who attend a review session to be 86%. At the 0.05 level of significance, attendance of the review session is associated with a higher exam score. The review session is helpful to students at the 0.01 level of significance. 64.According to a recent poll, 40.5% of people aged 25 years or older in the state of Massachusetts have a bachelor’s degree or higher. The poll also reported that 30.0% of people aged 25 years or older in the state of Delaware have a bachelor’s degree or higher. The poll sampled 354 residents of Massachusetts and 210 residents of Delaware. The data was calculated to be significant at the 0.013 level. Determine the meaning of this significance level. Select the correct answer below: At the 0.013 level of significance, a larger percentage of residents from Massachusetts have bachelor’s degrees. It is not unusual to see 30.0% of a sample of 210 residents of Delaware have bachelor’s degrees because level of education varies. It is certain that more residents of Massachusetts have bachelor’s degrees than do residents of Delaware. We can expect about 40.5% of any group of 354 Massachusetts residents to have a bachelor’s degree or higher. 65. A survey was conducted to see whether age has an association with the belief that a master’s degree or higher provides an advantage in one’s career. Of the 524 adults between the ages of 22 and 25 surveyed, 56% believed that a master’s degree has value in a person’s career path. Of the 458 adults surveyed between the ages of 40 and 45, 52% also believed that a master’s degree has value in a person’s career path. Assuming age is not associated with this belief, the probability of the data being the result of chance is calculated to be 0.21. Interpret this calculation. Select the correct answer below: We can expect 56% of all adults between the ages of 22 and 25 to believe a master’s degree or higher provides an advantage in one’s career. The data is statistically significant at the 0.05 level of significance in showing that age has an association with the belief that a master’s degree or higher provides an advantage in one’s career. Age does not have any association with the belief that a master’s degree or higher provides an advantage in one’s career. The data are not statistically significant at the 0.05 level of significance in showing that age has an association with the belief that a master’s degree or higher provides an advantage in one’s career. 66. A farmer claims that the average mass of an apple grown in his orchard is 100g. To test this claim, he measures the mass of 150 apples that are grown in his orchard and determines the average mass per apple to be 98g. The results are calculated to be statistically significant at the 0.01 level. What is the correct interpretation of this calculation? Select the correct answer below: The data are not statistically significant at the 0.05 level. The mean mass of any 150 apples grown in the farmer's orchard is 98g. At the 0.01 level of significance, the mean mass of the apples grown in the farmer's orchard is different from 100g. At the 0.01 level of significance, the mean mass of the apples grown in the farmer's orchard is 98g.