Answers to All Exercises

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1 CAPER 10 CAPER 10 CAPER10 CAPER REFRESING YOUR SKILLS FOR CAPER 10 1a b c a b. 7 is most likely; probability of 7 is c a b c a b c ANSWERS O ALL EXERCISES

2 LESSON a. 6 0.; ; b. eperimental a b c d e. theoretical 3a b c d e a b c a. eperimental 5b. theoretical 5c. eperimental 6a. Answers will vary. 6b. Possible answer: Use the random integer com mand on the calculator to simulate rolling a die. 6c. Answers will vary. 6d. Answers will vary. Sum your answers from 6c and divide the answer by 10. 6e. Answers will vary. Long-run averages should tend toward 6 turns in order to roll a Answers will vary. Each of these methods has shortcomings. 7. i. Middle numbers (3 7) are more common than getting only 1 or or or 9 heads in one trial of dropping pennies. 7. ii. Very few pencils will be at 0 or 1 in.; students throw away their pencils long before that. 7. iii. his is the best method, although books tend to open to pages that are used more than others. a. Answers will vary. b. he long-run eperimental probability should show that 1 6 of all rolls are a 3. c. Answers will vary. he points should level out to a straight line at y If you considered 5 s instead of 3 s, the data should level out to the same value. d. Answers will vary but should be close to 1 6. e. P(3) here are si equally likely outcomes, and 3 is one of them, so the theoretical probability is a. ; b. 5; c. 10; d. ; e. 10; a., or b. to a. 1 square units 11b. square units 11c. 11d e f. 0; 0 1a. y 6 1b. y 0 1c a b c d a b c d a. 53 pm, at point C 15b. 0 pm, at point A, the nucleus ANSWERS O ALL EXERCISES 115

3 15c. he probability starts at 0 at the nucleus, increases and peaks at a distance of 53 pm, and then decreases quickly, then more slowly, but never reaches y 6 y y 3 y 17. log ac b a. y 1 (, 9) 5 y 6 3 y 15 A P 1 ( 6, 1) B (6, 3) 6y 1 C 19b. (, 9), ( 6, 1), (6, 3) 19c. 6 units 0. a parabola with focus (3, 0) and directri y 6; y a. Set i should have a larger standard deviation because the values are more spread out. 1b. i. 35, s.3 1b. ii. 117, s 3.5 1c. he original values of and s are multiplied by 10. 1c. i. 350, s 3.5 1c. ii. 1170, s 35. 1d. he original values of are increased by 10, and the original values of s are unchanged. 1d. i. 5, s.3 1d. ii. 17, s ANSWERS O ALL EXERCISES

4 1. e 1 e e 3 v 1 v v 1 v v 1 v LESSON 10.. P(a) 0.675; P(b) 0.075; P(c) 0.05; P(d) 0.; 1 d 1 d1 d1 d1 d1 d1 3. P(a) 0.7; P(b) 0.3; P(c) 0.1; P(d) 0.; P(e) 0.; P(f) 0.; P(g) a b c d d d d d d 5a. he probability of selecting a junior given that a sophomore has already been selected; P(J S1) 5b c a ; ; ; b. No, because the probabilities of the four paths are not all the same 6c a. 7b c d e f a. b. c. 9a. 9b. 9c. 16 9d. 3 9e. 10 9f. n 10a. 1 10b. 1 10c. independent 10d. Both statements reveal misconceptions about the probability of independent events. he probability of heads is always 1 ; the coin does not remember how it landed previously. 11a b c d e f. 1 11g ANSWERS O ALL EXERCISES 117

5 1a M M M D 0.01 G 0.19 D 0.0 G 0.3 D G b c d a b c d a. the probability that the roll is odd and it is a 3 or a 5; b. the probability that the roll is a 3 or a 5 given that the roll is odd; c. the probability that the roll is odd given that it is a 3 or a 5; 1 16a b c. he events are dependent, because P(10th grade female) P(10th grade). he probability of choosing a 10th grader from the female students is greater than the probability of choosing a 10th grade student a. 3 i 1b. i 1c i 19. P(orange) 0.15; P(blue) a b ANSWERS O ALL EXERCISES

6 LESSON % of the students are sophomores and not in advanced algebra. 15% are sophomores in advanced algebra. 1% are in advanced algebra but are not sophomores. 63% are neither sophomores nor in advanced algebra. a. 0.5 b. 0.1 c d Sophomore In advanced algebra 9. B A Sophomore In advanced algebra No. P(S) P(A) , P(S and A) hese must be equal if the events are independent. 5a. yes, because they do not overlap 5b. No. P(A and B) 0. his would be the same as P(A) P(B) if they were independent. 6a. French Music 6b. approimately 13% 6c. 7 7a. A B 0. 7b. i 0.0 7b. ii b. iii P(A and B) 0., 0.5 P(A or B) 0.9. he first dia gram shows P(A and B) 0 and P(A or B) 0.9. he second dia gram shows P(A and B) 0. and P(A or B) 0.5. A B 10a. yellow 10b. cyan 10c. white 10d. blue 10e. green 10f. black 11a. Amber Carol Bob 11b c. 0. 1a b c d approimately a. 3 15b c. y 15y Answers will vary. Making a free throw is not random as flipping a coin is, so Janie may be improving. owever, even if her overall accuracy is still 50%, she could have a sequence of 5 successes in a row. ANSWERS O ALL EXERCISES 119

7 LESSON 10. 1a. Yes; the number of children will be an integer, and it is based on a random process. 1b. No; the length may be a non-integer. 1c. Yes; there will be an integer number of pieces of mail, and it is based on random processes of who sends mail when. a. Yes; the result of each call is independent of other calls, and you stop counting when she is successful. b. No; the number of cats is a discrete random variable, but you don t stop counting when you get the first cat; it is not geometric. c. No; you are counting minu tes until you hear a song, but because not all songs are the same length and minutes are not equivalent to songs, you are working with two different types of variables. 3a P() b. 1.5 a. approimately 0.06 b. approimately a. 0 5b. 0 6a. Answers will vary. heoreti cally, after 10 games Sly should get about 3 points, and Andy should get about 1. 6b. Answers will vary. heoreti cally, it should be close to c. 15 Andy gets Sly gets. 6d e. Answers will vary. One possible answer is 5 points for Sly if the sum of the dice is less than and 7 points for Andy if the sum of the dice is greater than 7. 7a. $5 7b c. approimately $.33 a. Answers will vary. b. Sample answer: Assign each of the letters in the word CAMPION a different number from 1 to. Randomly generate numbers between 1 and. Count how many digits you must gene rate until you have at least one of each number. c. Answers will vary. d. Answers will vary. heoretically, it should be about boes. e. Answers will vary. he average number of boes should be about. 9a. 0. 9b = c. Successful hits Probability d. P(n) = 0.(0.) n 9e. geometric 6 9f. P(n 6) 1 0. (0.) i 0.10 i 0 10a points 10b. Answers will vary. 11a b. Number of defective radios 11c Probability P() P() ANSWERS O ALL EXERCISES

8 11d. On average, the engineer should epect to find 0.97 defective radio in a sample of approimately a. Metal Oval Small 15b. Calculated using the actual frequencies: Material Shape Oval 0.35 Size Small ri Large Oval ri n 5 log Metal Plas. Metal Plas. Metal Plas. Metal Plas. ANSWERS O ALL EXERCISES 11

9 LESSON a. Yes. Different arrangements of scoops are different. 1b. No. We are not counting different arrangements separately. 1c. No. Repetition is not allowed in permutations. 1d. No. Repetition is not allowed in permutations. a. 1 b. 7 c. n 1 d. n e. 1,0 f. n(n 1) g. n 1 3a. 10 3b. 500 (n )! 3c. 3d. n! a. b. 1 c. 56 d. 19 5a. 10,000; 7. 7 h 5b. 100,000; approimately d 5c n r 6, or n 6 and r 5, or n 10 and r 3, or n 70 and r 1 7. r factors. 60 9a. 0,30 9b c d. Sample answer: here are eight possible positions for Volume 5, all equally likely. So P(5 in rightmost slot) e. 0.5; sample answer: there are four evennumbered books that can be in the rightmost position out of the eight books. So the probability of an evennumbered book being on the right is f. 1 9g. 0,319 9h , Number of N permutations of N items ime s 10 3,6, s 1 79,001,600 min 13 6,7,00, h d ,100 yr 11a. 100,000 11b. 1,000,000,000 11c. 17,576,000 11d. 7,00,000 1a b. 000 min 33.3 h, or about 1 d 9 h 0 min 1c. about min, or about centuries 13a b c d. It is not possible because there are no brown-eyed (B) genes in the miture. 13e a b a b. You would epect to lose 0.50 point on each of the 10 tosses, or a total loss of 5 points. 1 16a b c a. y b. y 0.5( 5) 3 17c. y a. 1 1b. about 0.3 in 1 ANSWERS O ALL EXERCISES

10 LESSON a. 10 1b. 35 1c d. 1 a. 10 b. 35 c. 105 d. 1 3a. 7P! 7 C 3b. 7P 3 3! 7 C 3 3c. 7P! 7 C 3d. 7P 7 7! 7 C 7 3e. np r r! n C r. Neither; they are the same. 5. n 7 and r 3, or n 7 and r, or n 35 and r 1, or n 35 and r 3 6. r 6; 10!! 6! 10! ; the number of 10 things taken 6!! at a time is equal to the number of 10 things omitted 6 at a time. 7a. 35 7b Sample answer: In a true combination lock, the order in which the numbers are en tered would not matter. In com bination locks, the order of the numbers does matter, so they are more like permutation locks. owever, in a true permutation lock, repeated numbers would not be allowed. 9a. 9b. 9c. 16 9d. he sum of all possible combinations of n things is n ; a. approimately 3. yr 10b C 6 11a. 6 11b c d. n C n! (n )! 1a. 6,66,96,50 ways 1b. approimately ways 13a. 7C 3 50 C 13b. 7C 50 C c. 1 C C 13d. $3.0 1a. y y 1b. 3 3 y 3y y 3 1c. 3 y 6 y y 3 y speeds 16a is the probability that someone is healthy but tests positive. 16b. 0.0 is the probability that a healthy person tests positive. 16c is the probability that a person tests positive. 16d is the probability that a person who tests positive is healthy. 17. C 157, or C a. $6, b. 0 yr 11 mo 19a. 1, 1, or, 19b. 1, 3 ANSWERS O ALL EXERCISES 13

11 1a. 7 1b. 5,17,066, y 10 1c. 6,91,99 7 y 0 1d. 7 y 6 a b c. (0.5) n LESSON 10.7 d. approimately 0.6 3a b , c. 0.03, 0.50, d. Both the at most and at least numbers include the case of eactly. For eample, if eactly 5 birds (0.165) is subtracted from at least 5 birds (0.03), the result (0.03) is the same as ( at most 5 birds). 3e. he probability that at least 5 birds survive is 0.3%.. What is the probability of eactly 35 successes in 50 trials? 5. What is the probability of at least 35 successes in 50 trials? 6a.,,, 6b.,,, 6c. Both diagrams would look the same. First flip/coin Second flip/coin 6d. C 0 1 is the number of ways of getting 0 tails, C 1 is the number of ways of gett ing 1 tail, and C 1 is the number of ways of getting tails. 6e. here is 1 way of getting heads; there are ways of getting 1 head and 1 tail; and there is 1 way of getting tails. 7a. 3 y 6 y y 3 y 7b. p 5 5p q 10p 3 q 10p q 3 5pq q 5 7c d a. 50 C 0 p 10 q 0 or 50 C 10 p 10 q 0 b. the sum of terms 1 to 11, or 50 C r p r (1 p) 50 r c. P(r 10) d. no 10 r 0 9a. approimately b. approimately c. f () 30 C (0.97) 30 (0.03) 9d a b c a b c. 1. birds, or approimately birds 13. Answers will vary. his event will happen in 15.65% of trials. 1a b c d a. 1 3 Sum of the first terms Sum of all the terms b. f (10).59, f (100).705, f (1,000).717, f (10,000).71 15c. here is a long-run value of about ,70. Sample answer: Either the group of five students selected includes the new student or it doesn t. If the new student is included, then the other are selected from the remaining 5 class members, and this can be done 5 C 1,650 ways. If the new student is not selected, then all 5 are selected from the 5 original members, and this can be done 5 C 5 53,130 ways. his means there are 1,650 ways that the new student is part of the group and 53,130 ways that he or she is not. his makes 1,650 53,130 65,70 ways to select 5 students. 17a. he eperimental pro babilities are likely to be differ ent from 0.5 and 0.5. In this sample simulation, P() 0.6 and P() ANSWERS O ALL EXERCISES

12 17b. he eperimental probabilities should be closer to 0.5 and 0.5, but are likely to not yet be eact. In this sample simulation, P() 0.9 and P() (distance, log (period)) Log period y Distance (millions of miles) (log (distance), log (period)) 17c. You can model the thumbtack by defining randompick with more points up to choose from. For eample, randompick( U, U, U, D, D ) would make the eperimental probability of points up approimately cm 19a. (distance, period) Period 1,000 (log (distance), period) Period 1,000 y y Distance (millions of miles) Log period 3 1 y Log distance (log (distance), log (period)) is the most linear of the four, so use the median-median line to find an equation to fit these points; y ; log (period) 1.50 log (distance) 9.3; 10 log (period) log (distance) 9.3 ; period distance b. 31,35; 61,566; 9,535; errors may be due to rounding. For more accurate results, the a- and b-values found from regression can be stored in variables in the calculator. 19c. period distance 3 0a. A r, where A area and r radius 0b. V l w h, where l length, w width, and h height 0c. N k, where d the distance from the hinge d Log distance ANSWERS O ALL EXERCISES 15

13 CAPER 10 REVIEW 1. Answers will vary depending upon whether you interpret the problem to imply random decimal numbers (between 0 and 10, non-inclusive) or random integers (0 to 10, inclusive). o generate random decimal numbers, you might look at a random-number table and place the decimal point after the first digit in each group of numbers. Alternatively, you could use a calculator command, such as 10*rand(numrials) on the I-Nspire, or 10*rand on the I- Plus. o generate random integers, you might number 11 chips or slips of paper and randomly select one. Alternatively, you could use a calculator command, such as randint(0, 10) on the I-Nspire or the I- Plus. a, b c d a b units a. F F F F F FF F F F FF FF F F FFF F F FF F FF F F FFF FF F F FFF FFF F F FFFF b. c. Because the order in which the true and false answers occur doesn t matter, use combinations: C 3. d a. Plain 0.36 Veggie 0.17 Chili 0.7 P 0.6 R 0.11 M 0.7 P 0.6 R 0.11 M 0.7 P 0.6 R 0.11 M 0.7 5b c d a. See below. 6b c d e Cats Dogs Other 9. approimately a. 1 10b c. 93, 930a 1 b 9 6a. (Chapter 10 Review) 9th grade 10th grade 11th grade 1th grade otal Ice cream Whipped cream otal ANSWERS O ALL EXERCISES