Chapter-Quizzes-2903y86 (3)

Quiz 5.1A AP Statistics Name: -190503238500 The probability of rolling two six-sided dice and having the sum on the two dice equal 7 is 16 . Interpret this probability. You roll two dice six times. Are you guaranteed to get a sum of 7 once? Explain. To pass the time during a long drive, you and a friend are keeping track of the makes and models of cars that pass by in the other direction. At one point, you realize that among the last 20 cars, there hasn’t been a single Ford. (Currently, about 16% of cars sold in America are Fords). Your friend says, “The law of averages says that the next car is almost certain to be a Ford.” Explain to your friend what he doesn’t understand about probability. A bag contains 10 equally-sized tags numbered 0 to 9. You reach in and, without looking, pick 3 tags without replacement. We want to use simulation to estimate the probability that the sum of the 3 numbers is at least 18. Describe the simulation procedure below, then use the random number table on the next page to carry out 10 trials of you simulation and estimate the probability. Mark on or above each line of the table so that someone can clearly follow your method. Random number table for question 3. 128 15689 14227 06565 14374 13352 49367 81982 87209 129 36759 58984 68288 22913 18638 54303 00795 08727 130 69051 64817 87174 09517 84534 06489 87201 97245 131 05007 16632 81194 14873 04197 85576 45195 96565 a 207 Quiz 5.1B AP Statistics Name: -190503238500 The probability of flipping four coins and getting four “heads” is 161 . Interpret this probability. You flip four coins 32 times. Are you guaranteed to get four “heads” twice? Explain. You are playing a board game with some friends in which each turn begins with rolling two dice. In this game, rolling “doubles”—the same number on both dice—is especially beneficial. You’ve rolled doubles on your last three turns, and one of your friends says, “No way you’ll roll doubles this time, it would be nearly impossible.” Explain to your friend what he doesn’t seem to understand about probability. A school’s debate club has 10 members, 6 females and 4 males. If the team decides to pick two members randomly to participate in a debate, what is the probability that both of the chosen members are female? We want to use simulation to estimate this probability. Describe the simulation procedure below, then use the random number table on the next page to carry out 10 trials of your simulation and estimate the probability. Mark on or above each line of the table so that someone can clearly follow your method. 4244975-616204000 208 A Random number table for question 3. 141 96767 35964 23822 96012 94591 65194 50842 53372 142 72829 50232 97892 63408 77919 44575 24870 04178 143 88565 42628 17797 49376 61762 16953 88604 12724 144 62964 88145 83083 69453 46109 59505 69680 00900 Quiz 5.1C AP Statistics Name: -190503238500 A couple has two sons and decide to have a third child. The husband says, “We’re bound to have a daughter this time: things balance out.” The wife says, “Nonsense! Two boys in a row means we are more likely to have another boy.” Comment on this disagreement, based on your understanding of probability. The probability that a randomly selected person in the United States is left-handed is about 0.14. Use this probability to explain what the Law of Large Numbers says. Among the 28 students in Mr. Millar’s Calculus BC class, 8 are left-handed. Could this have happened by chance alone? Describe how you would use a random number table to simulate the proportion of left-handers in a class of 28 students if they were chosen randomly from a population that is 14% left-handed. Do not perform the simulation. Below are the number of left-handers in 100 simulated classes of 28 students, assuming that students are selected randomly from a population in which 14% of individuals are left-handed. What do these results suggest about the proportion of lefties in Mr. Millar’s class? 11861809969500 Quiz 5.2A AP Statistics Name: -190503238500 1. Suppose you choose a random U.S. resident over the age of 25. The table below is a probability model for the education level the selected person has attained, based on data from the American Community Survey from 2006-2008. Education level attained Probability No high school diploma 0.20 High School diploma or GED 0.22 Some college 0.29 Bachelor’s degree 0.19 Graduate or professional degree ? 1282700-1109345001285875-1111885003526155-1111885004440555-1111885001282700-921385001282700-733425001282700-551180001282700-370205001282700-188595001282700-635000 (a) What is the probability that a randomly selected person has a graduate or professional degree? (That is, fill in the space marked with a “?”) Show your work. (b) What is the probability that a randomly-selected person has at least a high school diploma? Show your work. 2. There are 35 students in Ms. Ortiz’s Precalculus class. One day, 24 students turned in their homework and 14 turned in test corrections. Eight of these students turned in both homework and test corrections. Suppose we randomly select a student from the class. (a) Fill in the Venn diagram below so that it describes the chance process involved here. Let H = the event “turned in homework” and C = the event “turned in corrections.” 1449070254000411607025400014439906985001443990142684500 What is the probability that a randomly-chosen student turned in neither homework nor corrections? Justify you answer with appropriate calculations. Below is a two-way table that describes responses of 120 subjects to a survey in which they were asked, “Do you exercise for at least 30 minutes four or more times per week?” and “What kind of vehicle do you drive?” Car type Total Sedan SUV Truck Exercise? Yes 25 15 12 52 No 20 24 24 68 Total 45 39 36 120 Suppose one person from this sample is randomly selected. List two mutually exclusive events for this chance process. What is the probability that the person selected drives an SUV? What is the probability that the person selected drives either a sedan or a truck? What is the probability that the person selected drives a truck or exercises four or more times per week? Quiz 5.2B AP Statistics Name: -190503238500 1. The table below is a probability model for the number of cars in a randomly-selected household in the United States. (Based on U.S. Census 2000 data). Number of cars 0 1 2 3 4 5 or more Probability 0.07 0.19 0.47 ? 0.06 0.02 320675-37020500323850-372745001421130-372745002161540-372745002901315-372745003642360-372745004382770-372745005122545-372745005863590-37274500320675-18859500320675-635000 (a) What is the probability that a randomly selected household has three cars? (That is, fill in the space marked with a “?”) Show your work. (b) What is the probability that a randomly-selected household has at least 2 cars? Show your work. 2. Last Saturday at Pasquale’s Pizzas and Wings, 60 customers were served over the course of the evening. Fifty-two customers ordered pizza and 16 ordered buffalo wings. Twelve of these customers ordered both pizza and wings. Suppose we select a customer from last Saturday at random. (a) Fill in the Venn diagram below so that it describes the chance process involved here. Let P = the event “ordered pizza” and W = the event “ordered wings.” 1535430393700042024303937000153035043815001530350146304000 What is the probability that a randomly-chosen customer did not order wings or pizza? Justify you answer with appropriate calculations. The table below gives the counts (in thousands) of earned degrees in the United States in a recent year, classified by level and by the gender of the degree recipient. Degree Total Bachelor’s Master’s Professional Doctoral Female 616 194 30 16 856 Male 529 171 44 26 770 Total 1145 365 74 42 1626 Suppose one degree recipient from this group is selected randomly. List two mutually exclusive events for this chance process. What is the probability that the person selected earned a Master’s degree? (c) What is the probability that the person selected earned a Professional or Doctoral degree? (d) What is the probability that the person selected is female or earned a Master’s degree? Quiz 5.2C AP Statistics Name: -190503238500 Suppose you toss one coin and roll one six-sided die. List the outcomes in the sample space. Find the probability of getting a head. Find the probability of getting a 1, 2, or 3 on the die. Find the probability of getting a head or a five. In a statistics class there are 18 juniors and 10 seniors; 6 of the seniors are females, and 12 of the juniors are males. If a student is selected at random, find the probability of selecting a junior or a female not a junior male Consolidated Builders has bid on two large construction contracts. The company president believes that the probability of winning the first contract (event A) is 0.6, that the probability of winning a second (event B) is 0.3, and that the probability of winning both jobs is 0.1. Construct either a Venn diagram or a two-way table that summarizes what you know about events A and B. What is P A or B —the probability that Consolidated wins at least one of the job? Write each of the following events in terms of A, B, Ac, and Bc, and use the information above to calculate the probability of each. Consolidated wins both jobs. Consolidated wins the first job but not the second. Either Consolidated does not win the first job or wins the second. Consolidated does not win either job. Quiz 5.3A AP Statistics Name: -190503238500 1. Ivy conducted a taste test for four different brands of chocolate chip cookies. Below is a two-way table that describes which cookie each subject preferred and their gender. Cookie Brand A B C D Totals Female 4 6 13 13 36 Male 22 11 11 14 58 Totals 26 17 24 27 94 Suppose one subject from this experiment is selected at random. Find the probability that the selected subject preferred Brand C. Find the probability that the selected subject preferred Brand C, given that she is female. Are the events “preferred Brand C” and “female” independent? Explain. Are the events “preferred Brand C” and “female” mutually exclusive? Explain. If a random sample of two subjects is selected, what is the probability that neither preferred Brand A? Officials at Dipstick College are interested in the relationship between participation in (interscholastic) sports and graduation rate. The following table summarizes the probabilities of several events when a male Dipstick student is randomly selected. Event Probability Student participates in sports 0.20 Student participates in sports and graduates 0.18 Student graduates, given no participation in sports 0.82 Find the probability that a student graduates, given that he participates in sports. Find the probability that the individual does not graduate, given that he participates in sports. Draw a tree diagram to summarize the given probabilities and those you determined above. Find the probability that the individual does not participate in sports, given that he graduates. Quiz 5.3B AP Statistics Name: -190503238500 What age groups use social networking sites? A recent study produced the following data about 768 individuals who were asked their age and which of three social networking sites they used most often. (People who did not use such sites were excluded from the study). Age Group (Years) Web site 0 – 24 25 – 44 45 – 64 Over 65 Totals Facebook 77 105 114 12 308 Twitter 46 110 81 7 244 LinkedIn 15 97 95 9 216 Totals 138 312 290 28 768 Suppose one subject from this study was selected at random. Find the probability that the selected subject preferred Twitter. Find the probability that the selected subject preferred Twitter, given that he or she was in the 45 – 64 age group. Are the events “preferred Twitter” and “age group 45 – 64” independent? Explain. Are the events “preferred Twitter” and “age group 45 – 64” mutually exclusive? Explain. If a random sample of two subjects were selected, what is the probability that neither preferred Twitter? Some days, Ramon drives to work. The rest of the time he rides his bike. Suppose we choose a random work day. The following table gives the probabilities of several events. Event Probability Drives to work 0.20 Drives and is late for work 0.05 Late for work, given he bikes 0.30 Find the probability that Ramon is late for work, given that he drives. Find the probability that Ramon is not late for work, given that he drives. Draw a tree diagram to summarize the given probabilities and those you determined above. (d) Find the probability that Ramon drove to work, given that he is late. Quiz 5.3C AP Statistics Name: -190503238500 Consider the following activity: The letters in the word AARDVARK are printed on identical plastic cards with one letter per card. The eight cards are then placed in a hat, and one card is randomly chosen (without looking) from the hat. The chance process we are interested in is what letter is on the selected card. List the sample space S of all possible outcomes. Make a table that shows the set of outcomes and the probability of each outcome: 2254251905000022860018732500117221018732500591566018732500 Outcome 2254259271000 Probability 2254259334500 (c) Consider the following events: V: the letter chosen is a vowel. F: the letter chosen falls in the first half of the alphabet (that is, between A and M). List the outcomes in each of the following events, and determine their probabilities: V = { P(V) = F = { P(F) = V or F = { P(V or F) = Fc = { P(Fc) = V and F = { P(V and F) = V given F = { P(V|F) = Are the events V and F are independent? Explain. Are the events V and F mutually exclusive? Explain. Suppose a person was having two surgeries performed at the same time by different operating teams. Assume (unrealistically) that the two operations are independent. If the chances of success for surgery A are 85%, and the chances of success for surgery B are 90%, what is the probability that both will fail? Parking for students at Central High School is very limited, and those who arrive late have to park illegally and take their chances at getting a ticket. Joey has determined that the probability that he has to park illegally and that he gets a parking ticket is 0.07. He recorded data last year and found that because of his perpetual tardiness, the probability that he will have to park illegally is 0.25. Suppose that Joey arrived late once again this morning and had to park in a no-parking zone. Can you find the probability that Joey will get a parking ticket? If so, do it. If not, explain what additional information is needed in order to find the probability. Chapter 5 Solutions Quiz 5.1A 1. (a) If two dice were rolled many, many times, the proportion of rolls that resulted in the sum on the dice equaling 7 would be about one sixth. (b) No. While we can predict the proportion of 7’s rolled in the long run, the proportion of 7’s rolled in the short run is unpredictable. 2. Assuming that the brand of each car is independent of other cars, the probability that the next car is a Ford does not change, regardless of what brands preceded it. Only in the very long run can we be sure that the number of Fords will approach whatever the expected probability is. 3. Assign digits 0 through 9 on the table to correspond to the numbers on the tags. Choose digits from the table until 3 digits without repeats are chosen. Add the three digits and determine if the sum is 18 or more. Do this 20 times and calculate the proportion of times that the sum is 18 or more. Starting at the beginning of the table provided, the 20 sums are: 12, 18, 13, 11, 12, 14, 8, 10, 16, 21. Estimate of probability is 2/10 = .20. Quiz 5.1B (a) If four coins were flipped many, many times, the proportion of times all four coins would come up “heads” would be about 1/16. (b) No. While we can predict the proportion of times we get 4 heads in the long run, in the short run that proportion is unpredictable. 2. Since dice rolls are independent, previous rolls have no impact on the probability of the next roll. Only in the very long run can we be confident that the proportion of doubles will approach the expected probability. 3. Assign the digits 0 through 5 to female club members and 6 through 9 to male club members. Choose two numbers from the random digits table, ignoring repeats, and determine the gender of the two club members chosen. Do this 20 times and calculate the proportion of times both people are female. Starting at the beginning of the table provided, the genders of the first 20 sets of two are: MM, MM, MF, FM, MF, FF, MF, FM, MF, FF. Estimate of probability is 2/10 = .20. Quiz 5.1C 1. Neither is correct. The probability of having a male child is not influenced by the genders of previous offspring. Only in the long run can we expect the proportion of male children to approach the expected probability (about 51%, as it turns out). 2. (a) As we randomly select more and more people, the proportion of left-handed people will get closer and closer to 0.14. (b) Assign 01 through 14 to left-handers and 15 through 00 to right-handers. Choose 28 two-numbers from the random digits table and count the number of left-handers in the group. Do this many, many times. (c) The number of left-handers is equal or greater than 8 in only 4 of 100 simulated classes of 28 students. This suggests that the number of lefties in Mr. Millar’s class is unusual. Quiz 5.2A 1. (a) Since all the probabilities in the sample space must add up to 1, P(Graduate or professional degree) = 1 – 0.90 = 0.10. (b) One possible method: using the complement rule, this is 1 – P(No high school diploma), or 1 – 0.20 = 0.80. 2. (a) Venn diagram at right. 24 14 8 30 1 (b) 1 P H C 1 1 0.143 35 35 35 35 7 3. (a) “Yes” and “No,” or any pair of car types. P SUV 12039 1340 0.325 P sedan or truck 12045 12036 12081 2740 0.675 P truck or yes 12036 12052 12012 12076 1930 0.633 907415-847090001494790-453390001875790-453390002272665-453390001357630-59690001738630-59690002117090-59690002515235-5969000 Quiz 5.2B H C 16 8 6 35 35 35 -299085-586740005 35 (a) Since all the probabilities in the sample space must add up to 1, P(3 cars) = 1 – 0.81 = (b) One possible method: using addition of disjoint events: 0.47 + 0.19 + 0.06 + 0.02 = 0.74. (a) Venn diagram at right. 4234815381000 52 16 12 56 1 (b) 1 P P W 1 1 0.067 60 60 60 60 15 3. (a) “Female” and “Male,” or any pair of degree types. P Master's 1626365 0.224 P Professional or Doctoral 162674 162642 1626116 81358 0.071 P female or Master's 1626856 1626365 1626194 10271626 0.632 1174115-846455002153920-453390002611120-453390003559175-453390001791335-59055002248535-5905500 P W 40 12 4 60 60 60 4 60 Quiz 5.2C 6 1 1. (a) S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} (b) P head or 2 12 (c) P 1 or 2 or 3 2 2 2 6 or 1 (d) P head or five 6 2 1 7 12 2 12 12 12 12 12 12 12 2. (a) P junior or female 18 12 6 24 6 0.857 28 28 28 28 7 12 16 4 0.571 (b) P not a junior male 1 P junior and male 1 28 28 7 3. (a) Event B A B Y N 0.5 0.1 0.2 Y Event A 0.1 0.5 0.2 N 0.2 0.2 (b) P A B 0.6 0.3 0.1 0.8 0 (c) i.) A B or (A and B); 0.1 ii.) A BC or (A and BC); 0.5 [ A B C iii.) AC B or (AC or B); 0.5 iv.) AC BC or (AC and BC); 0.2 is also correct.] 735330-175514000 Quiz 5.3A 24 13 1. (a) P C 0.255 (b) P C | F 0.361 94 36 No, From parts (a) and (b), P C | F P C No. The occurrence of one event does not preclude the occurrence of the other; it’s possible that a subject prefers Brand C and is also female. That is P (C F ) 0 . (e) 68 subjects did not prefer brand A, so P AC AC 9468 6793 0.521 2. See tree diagram for event names. (a) P G | S P S G .18 0.9 P S .20 (b) P NG | S 1 P G | S 1 0.9 0.1 (c) Tree diagram at right. (d) P ( N | G) P N G .8 .82 0.785 P G .8 .82 .2 .9 .9 G .2 S .1 NG .82 G .8 N -151130-126492000 .18 NG Quiz 5.3B 1. (a) P Twitter 244 0.318 768 81 (b) P Twitter | 45-64 0.279 (c) No. From parts (a) and 290 (b), P Twitter | 45 64 P Twitter (d) No. The occurrence of one event does not preclude the .2 occurrence of the other; it’s possible that a subject preferred Twitter and is also in the 45 – 64 age group. That is, P(Twitter 45 64) 0 .8 4284980-70866000 (e) 524 subjects did not prefer Twitter, so P Not Twitter Not Twitter 524768 767523 0.465 2. See tree diagram for event names. (a) P L | D ..2005 0.25 3276600-13970000 (b) P NL | D 1 P L | D 1 0.25 0.75 (c) Tree diagram at right. (d) P ( D | L) P D L .2 .25 P L .2 .25 .8 .3 .25 L D .75 NL .3 L B 340995-35560000352425-126555500 .7 NL 0.172 Quiz 5.3C 1. (a) S = {A, D, K, R, V} (b) Outcome A D K R V Probability 3/8 1/8 1/8 2/8 1/8 (c) V = {A}, P(V) = 3/8; F = {A, D, K}, P(F) = 5/8; V or F = {A, D, K}, P(V or F) = 5/8; Fc = {R, V}, P(Fc) = 3/8; V and F = {A}, P(V and F) = 3/8; V given F = {A}, P(V|F) = 3/5. (d) No, since P (V ) P (V | F ) (e) No, since P (V and F)= 83 0 . 2. P AC B C P AC P B C 1 P A 1 P B 1 .85 1 .9 .015 3. P Ticket | Illegal Park P (Illegal park Ticket) 0.07 0.28 P (Illegal Park) 0.25