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MTH101_Chap8 - u 23 2 How would you interpret P(E = 1 How...

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Unformatted text preview: u. 23. 2... How would you interpret P(E) = 1? How would you interpret P(E) = 0? A fair die is painted so that 3 sides are red, 2 sides are white, and] side is blue. The die is rolled once. In Problems 3—6, find the probability of the given event. The top side is blue. The top side is red or white. The top side is red, white, or blue. The top side is not white. The top side is blue or white. The top side is white. The top side is red and white. . The top side is green. Refer to the description of a standard deck of 52 cards and :Figure 4 on page 382. An experiment consists of drawing 1 ‘ card from a standard 52-card deck. In Problems 11—20, what is ' the probability of drawing A red card? A diamond? A face card? An ace or king? A black jack? The queen of hearts? A ten? A red face card? A two or spade? A club or diamond? In a family with 2 children, excluding multiple births, what is the probability of having 2 children of the opposite gender? Assume that a girl is as likely as a boy at each birth. In a family with ’2 children, excluding multiple births, what is the probability of having 2 girls? Assume that a girl is as likely as a boy at each birth. A store carries four brands of CD players, J, G, P, and S. From past records, the manager found that the relative frequency of brand choice among customers varied. Which of the following probability assignments for a par- ticular customer choosing a particular brand of CD player would have to be rejected? Why? (A) P(J) = .15, P(G) = —.35,P(P) = .50, P(S) = .70 (B) P(J) = .32, P(G) = .28, HP) = .24, P(S) = .30 (C) P(J) = .26, P(G) = .14, P(P) = .30, P(S) = .30 Using the probability assignments in Problem 23C, what is the probability that a random customer will not choose brand S? ' S e c ti o n 8 — 1 Sample Spaces, Events. and Probability 403 25. Using the probability assignments in Problem 23C. what is the probability that a random customer will choose brand I or brand P7 26. Using the probability assignments in Problem 23C, what is the probability that a random customer will not choose brand I or brand P? 27. In a family with 3 children, excluding multiple births. what is the probability of having 2 boys and 1 girl, in that order? Assume that a boy is as likely as a girl at each birth. 28. In a family with 3 children, excluding multiple births. what is the probability of having 2 boys and 1 girl, in any order? Assume that a boy is as likely as a girl at each birth. 29. A small combination lock on a suitcase has 3 wheels, each labeled with the 10 digits from 0 to 9. If an opening com- bination is a particular sequence of 3 digits with no repeats, what is the probability of a person guessing the right combination? ’ 30. A combination lock has 5 wheels, each labeled with the 10 digits from 0 to 9. If an opening combination is a par- ticular sequence of 5 digits with no repeats, what is the probability of a person guessing the right combination? Refer to the description of a standard deck of 52 cards and Figure 4 on page 382. An experiment consists of dealing 5 cards from a standard 52-card deck. In Problems 31 -—34, what is the probability of being dealt 31. 5 black cards? 32. 5 hearts? 33. 5 face cards? 34. 5 nonface cards? 35. 'I‘Wenty thousand students are emailed at a state university. A student is selected at random and his or her birthday (month and day, not year) is recorded. Describe an appropriate sample space for this experiment, and assign acceptable probabilities to the simple events. What are your assumptions in making this assignment? 36. 'In a hotly contested three-way race for the US. Senate, polls indicate the two leading candidates are running neck-and-neck while the third candidate is receiving half the support of either of the others Registered voters are chosen at random and are asked which of the three will get their vote. Describe an appropriate sample space for this random survey experiment and assign acceptable probabilities to the simple events 37. Suppose that 5 thank-you notes are written and 5 envelopes are addressed. Accidentally, the notes are ran- domly inserted into the envelopes and mailed without checking the addresses What is the probability that all the notes will be inserted into the correct envelopes? 38. Suppose that 6 people check their coats in a checkroom. If all claim checks are lost and the 6 coats are randomly returned, what is the probability that all the people will get their own coats back? rm .4“, syn-w.» wag-'vwcevw: at; ‘ 404 CHAPTER 8 Probability An experiment consists of rotting twa fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 395) and assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 39—54. 39. Sum is 2. 40. Sum is 10. 41. Sum' 15 6. 42. Sum is 8. f43‘?Sum is less than 5.44. Sum is greater than 8. C 45. Sum' 13 not 7 or 11. 46. Sum is not 2, 4, or 6. {47. Sum 15 1.48. Sum is 13. 49, Sum 1s divisible by 3. 50. Sum is divisible by 4. 51. Sum 1s 7 or 11 (a “natural”). 52. Sum 15 2, 3, or 12 (“craps”). 53. Sum is divisible by 2 or 3. 54. Sum is divisible by 2 and 3. An experiment consists of tossing three fair (not weighted) coins, except one of the three coins has a head on both sides. Compute the probability of obtaining the indicated results in Problems 55—60. 55. 1 head 56. 2 heads 57. 3 heads 58. 0 heads 59. More than 1 head 60. More than 1 tail In Problems 61 —66, a sample space S is described Would it be reasonable to make the equally likely assumption? Explain. 61. A single card is drawn from a standard deck. We are interested in whether or not the card drawn is a heart, so an appropriate sample space is S = [H, N]. A single fair coin is tossed. We are interested in whether the coin falls heads or tails, so an appropriate sample space isS = [H, T}. A single fair die is rolled. We are interested in whether or not the number rolled is even or odd, so an appropriate sample space is S = {E, 0}. A nickel and dime are tossed. We are interested in the number of heads that appear, so an appropriate sample space is S = [0, 1, 2}. A wheel of fortune has seven sectors of equal area colored red, orange, yellow, green, blue, indigo, and violet. We are interested in the color that the pointer indicates when the wheel stops, so an appropriate sample space is S = {R, 0, Y, G, B. I, V}. A wheel of fortune has seven sectors of equal area colored red, orange, yellow, red, orange. yellow, and red. We are interested in the color that the pointer indicates when the wheel stops, so an appropriate sample space is = (R, 0, Y}. (A) Is it possible to get 19 heads in 20 flips of a fair coin? Explain. (B) If you flipped a coin 40 times and got 37 heads, would you suspect that the coin was unfair? Why or why not? If you suspect an unfair coin, what empirical probabilities would you assign to the simple events of the sample space? 62. 63. 65. 67. 68. (A) Is it possible to get 7 double 6's in 10 rolls of a pai fair dice? Explain (B) If you rolled a pair of dice 36 times and got 11 douh 6’s, would you suspect that the dice were unfair? Why or why not? If you suspect loaded dice, what empirical probability would you assign to the even of rolling a double 6? An experiment consists of rolling two fair ( not weighted) die, and adding the dots on the two sides facing up. Each die has ' ' number 1 on two opposite faces, the number 2 on two opposite“ faces, and the number 3 on two opposite faces Compute the probability of obtaining the indicated sums in Problems 69—7 69. 2 70. 3 71. 4 72. 5‘ 73. 6 74. 7 75. An odd sum 76. An even sum Refer to the description of a standard deck of 52 cards on pag 382. An experiment consists of dealing 5 cards from a standar; 52-card deck. In Problems 77—84, what is the probability of. being dealt 77. 78. 79. 80. 81. 82. 83. 5 face cards or aces? 5 numbered cards (2 through 10)? 4 aces? Four of a kind (4 queens, 4 kings, and so on)? A 10, jack, queen, king, and ace, all in the same suit? ' A 2, 3, 4, 5, and 6, all in the same suit? 2 aces and 3 queens? 84. 2 kings and 3 aces? In Problems 85—88, several experiments are simulated usingL the random number feature on a graphing calculator. For , example, the roll of a fair die can be simulated by selecting a ‘ random integer from 1 to 6, and 50 rolls of a fair die by selects ing 50 random integers from 1 to 6 (see Fig. A for Problem and your user’s manual). 85. From the statistical plot of the outcomes of rolling a f die 50 times (see Fig. B), we see, for example, that 1: number 4 was rolled exactly 11 times Figure for 85 (A) What is the empirical probability that the numbe~ 6 was rolled? (B) What is the probability that a 6 is rolled under“ the equally likely assumption? (C) Use a graphing calculator to simulate 100 rolls 0 a fair die and determine the empirical probabili ties of the six outcomes 89. Consumer testing. Twelve popular brands of beer are to S e C t i 0 n 8 — 2 Union, Intersection, and Complement of Events; Odds 405 86. Use a graphing calculator to simulate 200 tosses of a ‘ nickel and Guns, representing the outcomes HH. HT.TH, and TT by 1, 2, 3, and 4, respectively. (A) Find the empirical probabilities of the four outcomes (B) What is the probability of each outcome under the equally likely assumption? i i 37, (A) Explain how a graphing calculator can be used to i simulate 500 tosses of a coin. ; i i (B) Carry out the simulation and find the empirical probabilities of the two outcomes iplications t be used in a blind taste study for consumer recognition. (A) If 4 distinct brands are chosen at random from the 12 and if a consumer is not allowed to repeat any answers, what is the probability that all 4 brands could be identified by just guessing? (B) If repeats are allowed in the 4 brands chosen at ran- dom from the 12 and if the consumer is allowed to repeat answers, what is the probability of correct identification of all 4 by just guessing? Consumer testing. Six popular brands of cola are to be used in a blind taste study for consumer recognition. (A) If 3 distinct brands are chosen at random from the 6 and if a consumer is not allowed to repeat any answers, what is the probability that all 3 brands could be identified by just guessing? (B) If repeats are allowed in the 3 brands chosen at ran- dom from the 6 and if the consumer is allowed to repeat answers, what is the probability of correct identification of all 3 by just guessing? Personnel selection. Suppose that 6 female and 5 male applicants have been successfully screened for 5 posi- tions If the 5 positions are filled at random from the 11 finalists, what is the probability of selecting (A) 3 females and 2 males? (B) 4 females and 1 male? (C) 5 females? (D) At least 4 females? Committee selection. A 4-person grievance committee is to be composed of employees in 2 departments, A and B, with 15 and 20 employees, respectively. If the 4 people are (C) What is the probability of each outcome under the equally likely assumption? . From a box containing a dozen balls numbered 1 through 12. one ball is drawn at random. (A) Explain how a graphing calculator can be used to simulate 400 repetitions of this experiment. (B) Carry out the simulation and find the empirical probability of drawing the 8 ball. (C) What is the probability of drawing the 8 ball under the equally likely assumption? selected at random from the 35 employees, what is the probability of selecting (A) 3 fromA and 1 from B? (B) 2 from A and 2 from B? (C) All from A? (D) At least 3 from A? Medicine. A prospective laboratory technician is to be tested on identifying blood types from 8 standard classifi— cations (A) If 3 distinct samples are chosen at random from the 8 types and if the examinee is not allowed to repeat any answers, what is the probability that all 3 could be correctly identified by just guessing? (B) If repeats are allowed in the 3 blood types chosen at random from the 8 and if the examinee is allowed to repeat answers, what is the probability of correct _, identification of all 3 by just guessing? Medical research. Because of limited funds, 5 research outfits are to be chosen out of 8 suitable ones for a study on heart disease. If the selection is made at random, what is the probability that 5 particular research centers will be chosen? 95. Membership selection. A town council has 11 members, 6 Democrats and 5 Republicans. (A) If the president and vice-president are selected at random, what is the probability that they are both Democrats? (B) If a 3~person committee is selected at random, what is the probability that Republicans make up the majority? Section 8-2 UNION, INTERSECTION, AND COMPLEMENT OF EVENTS: ODDS I Union and Intersection I Complement of an Event I Odds I Applications to Empirical Probability S e c t i o n 8 — 2 Union, Intersection, and Complement of Events; Odds 415 In either case. :5 oddsforE=———— or 9:1 P(E') (B) The event that a person has tried one cola but not both is the event that the per- son has tried diet and not regular cola or has tried regular and not diet cola. In terms of sets, this is event E = (D F) R') U (D’ n R). Since D D R’ and D’ n R are mutually exclusive (look at the Venn diagram in Fig. 7), P(E) = P[(D n R') U (D' n R)] = P(D n R’) + P(D' n R) Hm_2 2 .11 = .3 + .4 = .7 P(E‘) .3 3 d ' = ——-—— = — = — : o dsagamstE P(E) .7 7 or 37 _ MATCH ED PROBLEM 8 If a resident from the city in Example 8 is selected at random, what is the (empirical) probability that (A) The resident has not tried either cola? What are the (empirical) odds for this event? (B) The resident has tried the diet cola or has not tried the regular cola? What are the (empirical) odds against this event? ___...__ Answers to Matched Problems 1. (A) :1; (B) g 2. (A) 112 (B) {—8 3. (4147—0 z .336 4. .92 5. .016 6. (A) 5 :31 (B) $31 7. % z .455 s. (A) P(D’ n R’) = .1; odds for D’ r) R' '= g or 1 :9 (B) P(D U R') = .6; odds against DU R4 = gor 2:3 Exe rc i se 8 -2 Problems [—6 refer to the Venn diagram shown below for A single card is drawn from a standard 52-card deck. Let D be even“ A and 3 in an equally likely sample space S. Find each the event that the card drawn is a diamond, and let F be the ' ofthg indicated probabilities event that the card drawn is a face card. In Problems 7—18, find the indicated probabilities S 7. P(D) 8. P(F) 9. P(F’) 10. P(D') 11. P(D n F) 12. P(D’ n F) 1. P(A n B) 13. P(D U F) 2. P(A U B) 14. P(D’ U F) ~ g. )P(A' UB) 15. P(DnF') .4‘ P(A n B’) 16. P(D’ n F’) (gm/1 u B)’) 17. P(D u F’) it. P((A n B)’) 18- P(D’ U P) 415 C H A PT E R 8 Probability In a lottery game, a single ball is drawn at random from a con- 1 B Miner that contains 25 identical balls numbered from I through 25. In Problems 19-24, use equation (I) or (2). indi- cating which is used, to compute the probability that the num- ber drawn is.- 19. Less than 6 or greater than 19 20. Divisible by 4 or divisible by 7 21. Divisible by 3 or divisible by 4 22. Odd or greater than 15 23. Even or divisible by 5 24. Less than 12 or greater than 13 If the probability is .51 that a candidate wins the election, what is the probability that he loses? 26. If the probability is .03 that an automobile tire fails in km than 50,000 miles, what is the probability that the tire does not fail in 50,000 miles? In Problems 27—30, use the equally likely sample space in Example 2 and equation (1) or (2), indicating which is used, to compute the probability of the following events: 27. AsumofSoré 28. Asumof9or10 29. 'Ihe number on the first die is a 1 or the number on the second die is a 1. 30. The number on the first die is a 1 or the number on the second die ls less than 3. ( 31. Given the following probabilities for an event E, find the , odds for and against E: f; (A)3 5 (B) 2 (C) .4 (D) .55 f 32. Given the following probabilities for an event E, find the odds for and against E: ‘1 (A) g (13) i (C) .6 (D) .35 \l 33. Compute the probability of event E if the odds in favor of \\ E are { \(A) g (B) u (C)? (0)2—9 34. Foarmepute the probability of event E if the odds in favor of \‘.\1. (A) 5 (B) 5 (C)3 7 (13% In Problems 35—40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. 35. If the odds for E equal the odds against E’. then P(E) = 1. 36. Ifthe oddsforEarea:b,then theoddsagainstEareb :a. 37. If P(E) + P(F) = P(E U F) + P(E n F), then E and F are mutually exclusive events 38. The theoretical probability of an event is less than or equal to its empirical probability. 39. If E and F are complementary events, then E and F are mutually exclusive. 40. If E and F are mutually exclusive events, then E and F are complementary. In Problems 41—44, compute the odds in favor of obtaining . 41. A head in a single toss of a coin 42. A number divisible by 3 in a single roll of a die 43. At least 1 head when a single coin is tossed 3 times 44. 1 head when a single coin is tossed twice In Problems 45—48, compute the odds against obtaining 45. A number greater than 4 in a single roll of a die 46. 2 heads when a single coin is tossed twice 47. A 3 or an even number in a single roll of a die 48. An odd number or a number divisible by 3 in a single roll: of a die 49. (A) What are the odds for rolling a sum of 5 1n a singl roll of two fair dice? (B) If you bet $1 that a sum of S will turn up, what should the house pay (plus returning your $1 bet) if , a sum of 5 turns up for the game to be fair? 50. (A) What are the odds for rolling a sum of 10 1n a sing] roll of two fair dice? (B) If you bet $1 that a sum of 10 will turn up, what 5 ' should the house pay (plus returning your $1 bet) ' a sum of 10 turns up for the game to be fair? A pair of dice are rolled 1,000 times with the following fre- quencies of outcomes: Sum-f 234567891011 Money 1(1 so so 70110 150170 141121) so Use these frequencies to calculate the approximate empirical probabilities and odds for the events in Problems 51 and 52 61) (A) The sum is less than 4 or greater than 9. (B) The sum is even or exactly divisible by 5 2. (A) The sum is a prime number or is exactly divisible by: (B) The sum is an odd number or exactly divisible by 3 In Problems 53—56, a single card is drawn from a standard 52- card deck. Calculate the probability of and odds for each even 53. A face card or a club is drawn. 54. A king or a heart is drawn. 55. A black card or an ace is drawn. 56. A heart or a number less than 7 (count an ace as 1) is, drawn. < 57. What is the probability of getting at least 1 diamond in 5-card hand dealt from a standard 52-card deck? 58. What is the probability of getting at least 1 black card i113}: 7-card hand dealt from a standard 52-card deck? 59. What is the probability that a number selected at random from the first 1.000 positive integers is (exactly) divisible, by 6 or 8? ‘ 60. What is the probability that a number selected at random; from the first 600 positive integers is (exactly) divisible 6 or 9? 43-311 "’1 .2 F?" =‘".' “3:“ (1 ,2..,..n:» ‘:.':,V1." ~1‘.' 1.. 1.1151, 1; 2 - 2‘ .. . . '1‘. its... -;1. .4».er stem: , space S are related to each other in order for the follow— ing equation to hold: P(A u B u C) : P(A) + P(B) + P(C) ~ P(A n B) , Explain how the three events A. B, and C from a sample space S are related to each other in order for the follow- ing equation to hold: P(A u B u C) = P(A) + P(B) + P(C) , Show that the solution to the birthday problem in Example 5 can be written in the form P365." 365'l For a calculator that has a PM function, explain why this form may be better for direct evaluation than the other form used in the solution to Example 5.’Il’y direct evalua- tion of both forms on a calculator for n = 25. P(E) = 1 — . Many (but not all) calculators experience an overflow error when computing P365," for n > 39 and when com- puting 365". Explain how you would evaluate P(E) for a...
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