EXERCISES Section 2.4 (45-69).sync-conflict-20240920-170341-QYY3JQG

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The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions of individuals in the various ethnic group-blood group combinations.

2. 某个特定国家的人口由三个种族群体组成。每个个体属于四种主要血型中的一种。附带的联合概率表给出了各种种族群体-血型组合中个体的比例。

	O	A	B	AB
1	.082	.106	.008	.004
2	.135	.141	.018	.006
3	.215	.200	.065	.020

Suppose that an individual is randomly selected from the population, and define events by $A=\{$ type A selected $\},B=\{$ type B selected $\}$ , and $C=\{$ ethnic group 3 selected $\}$ .

假设从该人口中随机选择一个个体，并通过 $A=\{$ 选择A型 $\},B=\{$ 选择B型 $\}$ ，以及 $C=\{$ 选择第3种族群体 $\}$ 来定义事件。

1. Calculate $P\left(A\right),P\left(C\right)$ , and $P\left(A\cap C\right)$ .

2. 计算 $P\left(A\right),P\left(C\right)$ 和 $P\left(A\cap C\right)$ 。

3. Calculate both $P\left(A\mid C\right)$ and $P\left(C\mid A\right)$ , and explain in context what each of these probabilities represents.

4. 计算出 $P\left(A\mid C\right)$ 和 $P\left(C\mid A\right)$ ，并解释每个概率在上下文中的含义。

5. If the selected individual does not have type B blood, what is the probability that he or she is from ethnic group 1 ?

6. 如果被选中的个体没有B型血，那么他或她来自第1种族群体的概率是多少？

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7. Suppose an individual is randomly selected from the population of all adult males living in the United States. Let $A$ be the event that the selected individual is over $6\mathrm{ft}$ in height, and let $B$ be the event that the selected individual is a professional basketball player. Which do you think is larger, $P\left(A\mid B\right)$ or $P\left(B\mid A\right)$ ? Why?

8. 假设从所有居住在美国的成年男性人口中随机选择一个个体。令 $A$ 为被选中的个体身高超过 $6\mathrm{ft}$ 的事件，令 $B$ 为被选中的个体是一名职业篮球运动员的事件。你认为哪个更大， $P\left(A\mid B\right)$ 还是 $P\left(B\mid A\right)$ ？为什么？

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9. Return to the credit card scenario of Exercise 12 (Section 2.2), and let $C$ be the event that the selected student has an American Express card. In addition to $P\left(A\right)=.6$ , $P\left(B\right)=.4$ , and $P\left(A\cap \mathrm{B}\right)=.3$ , suppose that $P\left(C\right)=.2$ , $P\left(A\cap C\right)=.15,P\left(B\cap C\right)=.1$ , and $P\left(A\cap B\cap C\right)=.08$ .

返回练习12（第2.2节）中的信用卡场景，设 $C$ 为选中学生拥有美国运通卡的事件。除了 $P\left(A\right)=.6$ 、 $P\left(B\right)=.4$ 和 $P\left(A\cap \mathrm{B}\right)=.3$ 之外，假设 $P\left(C\right)=.2$ 、 $P\left(A\cap C\right)=.15,P\left(B\cap C\right)=.1$ 和 $P\left(A\cap B\cap C\right)=.08$ 。

1. What is the probability that the selected student has at least one of the three types of cards?

2. 选中学生在三种卡中至少拥有一种的概率是多少？

3. What is the probability that the selected student has both a Visa card and a MasterCard but not an American Express card?

4. 选中学生同时拥有Visa卡和MasterCard但不是美国运通卡的概率是多少？

5. Calculate and interpret $P\left(B\mid A\right)$ and also $P\left(A\mid B\right)$ .

6. 计算并解释 $P\left(B\mid A\right)$ 以及 $P\left(A\mid B\right)$ 。

7. If we learn that the selected student has an American Express card, what is the probability that she or he also has both a Visa card and a MasterCard?

8. 如果我们知道选中学生拥有美国运通卡，那么他或她也同时拥有Visa卡和MasterCard的概率是多少？

9. Given that the selected student has an American Express card, what is the probability that she or he has at least one of the other two types of cards?

10. 在选中学生拥有美国运通卡的情况下，他或她至少拥有其他两种卡中的一种的概率是多少？

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11. Reconsider the system defect situation described in Exercise 26 (Section 2.2).

12. 重新考虑练习26（第2.2节）中描述的系统缺陷情况。

13. Given that the system has a type 1 defect, what is the probability that it has a type 2 defect?

14. 在系统有一个类型1缺陷的情况下，它有一个类型2缺陷的概率是多少？

15. Given that the system has a type 1 defect, what is the probability that it has all three types of defects?

16. 在系统有一个类型1缺陷的情况下，它有三个类型缺陷的概率是多少？

17. Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect?

18. 在系统至少有一个类型缺陷的情况下，它恰好有一个类型缺陷的概率是多少？

19. Given that the system has both of the first two types of defects, what is the probability that it does not have the third type of defect?

20. 在系统有前两种类型的缺陷的情况下，它没有第三种类型缺陷的概率是多少？

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21. The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk.

22. 伴随的表格提供了在特定机场售货亭购买单杯咖啡的人选择的咖啡类型信息。

	Small	Medium	Large
Regular	14%	20%	26%
Decaf	20%	10%	10%

Consider randomly selecting such a coffee purchaser.

考虑随机选择这样的咖啡购买者。

1. What is the probability that the individual purchased a small cup? A cup of decaf coffee?

2. 个人购买小杯咖啡的概率是多少？购买脱咖啡因咖啡的概率是多少？

3. If we learn that the selected individual purchased a small cup, what now is the probability that he/she chose decaf coffee, and how would you interpret this probability?

4. 如果我们知道被选中的个体购买了小杯咖啡，那么他/她选择脱咖啡因咖啡的概率现在是多少，你如何解释这个概率？

5. If we learn that the selected individual purchased decaf, what now is the probability that a small size was selected, and how does this compare to the corresponding unconditional probability of (a)?

6. 如果我们知道被选中的个体购买了脱咖啡因咖啡，那么现在选择小杯的概率是多少，这如何与(a)中的无条件概率相比较？

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7. A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinations.

8. 一家百货商店销售三种尺寸（小号、中号和大号）的运动衬衫，三种图案（格子、印花和条纹）以及两种袖长（长袖和短袖）。伴随的表格给出了各种类别组合的衬衫销售比例。

Short-sleeved

Size	Pl	Pr	St
S	.04	.02	.05
M	.08	.07	.12
L	.03	.07	.08

Long-sleeved

Size	Pl	Pr	St
S	.03	.02	.03
M	.10	.05	.07
L	.04	.02	.08

1. What is the probability that the next shirt sold is a medium, long-sleeved, print shirt?

2. 下一件售出的衬衫是中号、长袖、印花衬衫的概率是多少？

3. What is the probability that the next shirt sold is a medium print shirt?

4. 下一件售出的衬衫是中号印花衬衫的概率是多少？

5. What is the probability that the next shirt sold is a short-sleeved shirt? A long-sleeved shirt?

6. 下一件售出的衬衫是短袖衬衫的概率是多少？长袖衬衫的概率是多少？

7. What is the probability that the size of the next shirt sold is medium? That the pattern of the next shirt sold is a print?

8. 下一件售出的衬衫尺寸为中号的概率是多少？图案为印花的概率是多少？

9. Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium?

10. 假设刚刚售出的衬衫是短袖格子衬衫，那么它的尺寸为中号的概率是多少？

11. Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? Long-sleeved?

12. 假设刚刚售出的衬衫是中号格子衬衫，那么它是短袖的概率是多少？长袖的概率是多少？

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13. According to a July 31, 2013, posting on cnn.com subsequent to the death of a child who bit into a peanut, a 2010 study in the journal Pediatrics found that $8\%$ of children younger than 18 in the United States have at least one food allergy. Among those with food allergies, about $39\%$ had a history of severe reaction.

14. 根据一篇2013年7月31日在cnn.com上的帖子，该帖子是关于一名儿童咬花生后死亡的报道，一篇2010年发表在《儿科》杂志上的研究发现 $8\%$ 美国小于18岁的儿童中至少有一种食物过敏。在食物过敏的儿童中，大约 $39\%$ 有严重反应的历史。

15. If a child younger than 18 is randomly selected, what is the probability that he or she has at least one food allergy and a history of severe reaction?

16. 如果随机选择一个18岁以下的孩子，他或她至少有一种食物过敏并且有严重反应史的几率是多少？

17. It was also reported that $30\%$ of those with an allergy in fact are allergic to multiple foods. If a child younger than 18 is randomly selected, what is the probability that he or she is allergic to multiple foods?

18. 报道还称 $30\%$ 的过敏者实际上对多种食物过敏。如果随机选择一个18岁以下的孩子，他或她过敏于多种食物的几率是多少？

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19. A system consists of two identical pumps, #1 and #2. If one pump fails, the system will still operate. However, because of the added strain, the remaining pump is now more likely to fail than was originally the case. That is, $r=P\left(\#2\text{fails}\mid \#1\text{fails}\right)>P\left(\#2\text{fails}\right)=q$ . If at least one pump fails by the end of the pump design life in $7\%$ of all systems and both pumps fail during that period in only $1\%$ , what is the probability that pump #1 will fail during the pump design life?

20. 一个系统由两个相同的泵组成，编号为 #1 和 2。如果一个泵失效，系统仍然可以运行。然而，由于增加了负荷，剩下的泵现在比原来更容易失效。也就是说 $r=P\left(\#2\text{fails}\mid \#1\text{fails}\right)>P\left(\#2\text{fails}\right)=q$ 。如果在泵的设计寿命结束时，所有系统中有 $7\%$ 的系统至少有一个泵失效，而在这个期间内两个泵都失效的只有 $1\%$ ，那么在泵的设计寿命期间泵 #1 失效的几率是多少？

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21. A certain shop repairs both audio and video components. Let $A$ denote the event that the next component brought in for repair is an audio component, and let $B$ be the event that the next component is a compact disc player (so the event $B$ is contained in $A$ ). Suppose that $P\left(A\right)=.6$ and $P\left(B\right)=.05$ . What is $P\left(B\mid A\right)$ ?

22. 某个商店修理音频和视频组件。设 $A$ 表示下一个送来修理的组件是音频组件的事件，设 $B$ 为下一个组件是光盘播放器的事件（因此事件 $B$ 包含在 $A$ 中）。假设 $P\left(A\right)=.6$ 和 $P\left(B\right)=.05$ 。那么 $P\left(B\mid A\right)$ 是多少？

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23. In Exercise $13,A_i=\{$ awarded project $i\}$ , for $i=1,2,3$ . Use the probabilities given there to compute the following probabilities, and explain in words the meaning of each one.

24. 在练习 $13,A_i=\{$ 获奖项目 $i\}$ 中，对于 $i=1,2,3$ 。使用那里给出的概率计算以下概率，并用文字解释每个概率的意义。

25. $P\left(A_2\mid A_1\right)$ b. $P\left(A_2\cap A_3\mid A_1\right)$

26. $P\left(A_2\mid A_1\right)$ b. $P\left(A_2\cap A_3\mid A_1\right)$

27. $P\left(A_2\cup A_3\mid A_1\right)$ d. $P\left(A_1\cap A_2\cap A_3\mid A_1\cup A_2\cup A_3\right)$

28. $P\left(A_2\cup A_3\mid A_1\right)$ d. $P\left(A_1\cap A_2\cap A_3\mid A_1\cup A_2\cup A_3\right)$

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29. Deer ticks can be carriers of either Lyme disease or human granulocytic ehrlichiosis (HGE). Based on a recent study, suppose that $16\%$ of all ticks in a certain location carry Lyme disease, $10\%$ carry HGE, and $10\%$ of the ticks that carry at least one of these diseases in fact carry both of them. If a randomly selected tick is found to have carried HGE, what is the probability that the selected tick is also a carrier of Lyme disease?

30. 鹿蜱可以是莱姆病或人类粒细胞无形体病（HGE）的携带者。基于最近的一项研究，假设在某个特定地点的蜱虫中有 $16\%$ 携带莱姆病， $10\%$ 携带HGE，并且 $10\%$ 的蜱虫实际上同时携带这两种疾病。如果随机选取的蜱虫被发现携带了HGE，那么选取的蜱虫也是莱姆病的携带者的概率是多少？

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31. For any events $A$ and $B$ with $P\left(B\right)>0$ , show that $P\left(A\mid B\right)+P\left(A^{\prime }\mid B\right)=1.$

32. 对于任意事件 $A$ 和 $B$ ，如果 $P\left(B\right)>0$ ，证明 $P\left(A\mid B\right)+P\left(A^{\prime }\mid B\right)=1.$ 。

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33. If $P\left(B\mid A\right)>P\left(B\right)$ , show that $P\left(B^{\prime }\mid A\right)<P\left(B^{\prime }\right)$ . [Hint: Add $P\left(B^{\prime }\mid A\right)$ to both sides of the given inequality and then use the result of Exercise 56.]

34. 如果 $P\left(B\mid A\right)>P\left(B\right)$ ，证明 $P\left(B^{\prime }\mid A\right)<P\left(B^{\prime }\right)$ 。[提示：在给定不等式两边加上 $P\left(B^{\prime }\mid A\right)$ ，然后使用练习56的结果。]

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35. Show that for any three events $A,B$ , and $C$ with $P\left(C\right)>0$ , $P\left(A\cup B\mid C\right)=P\left(A\mid C\right)+P\left(B\mid C\right)-P\left(A\cap B\mid C\right).$

36. 证明对于任意三个事件 $A,B$ 、 $C$ 和 $P\left(C\right)>0$ ，如果 $P\left(A\cup B\mid C\right)=P\left(A\mid C\right)+P\left(B\mid C\right)-P\left(A\cap B\mid C\right).$ ，那么 。

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37. At a certain gas station, $40\%$ of the customers use regular gas $\left(A_1\right),35\%$ use plus gas $\left(A_2\right)$ , and $25\%$ use premium $\left(A_3\right)$ . Of those customers using regular gas, only $30\%$ fill their tanks (event $B$ ). Of those customers using plus, $60\%$ fill their tanks, whereas of those using premium, $50\%$ fill their tanks.

38. 在某个加油站， $40\%$ 的顾客使用普通汽油， $\left(A_1\right),35\%$ 使用高级汽油， $\left(A_2\right)$ 使用加铅汽油， $25\%$ 使用优质汽油。在那些使用普通汽油的顾客中，只有 $\left(A_3\right)$ 把油箱加满（事件 $30\%$ ）。在那些使用高级汽油的顾客中， $B$ 把油箱加满，而在那些使用加铅汽油的顾客中， $60\%$ 把油箱加满。

39. What is the probability that the next customer will request plus gas and fill the tank $\left(A_2\cap B\right)$ ?

40. 下一个顾客请求高级汽油并加满油箱的概率是多少 $\left(A_2\cap B\right)$ ？

41. What is the probability that the next customer fills the tank?

42. 下一个顾客加满油箱的概率是多少？

43. If the next customer fills the tank, what is the probability that regular gas is requested? Plus? Premium?

44. 如果下一个顾客加满了油箱，那么请求普通汽油、高级汽油和优质汽油的概率分别是多少？

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45. Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, $60\%$ have an emergency locator, whereas $90\%$ of the aircraft not discovered do not have such a locator. Suppose a light aircraft has disappeared.

46. 在某个国家，70% 在飞行中失踪的轻型飞机最终被找到。在找到的飞机中， $60\%$ 配备了紧急定位器，而在未找到的飞机中， $90\%$ 没有这样的定位器。假设一架轻型飞机已经失踪。

47. If it has an emergency locator, what is the probability that it will not be discovered?

48. 如果它配备了紧急定位器，那么它不被发现的概率是多少？

49. If it does not have an emergency locator, what is the probability that it will be discovered?

50. 如果它没有配备紧急定位器，那么它被发现的概率是多少？

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51. Components of a certain type are shipped to a supplier in batches of ten. Suppose that $50\%$ of all such batches contain no defective components, $30\%$ contain one defective component, and $20\%$ contain two defective components. Two components from a batch are randomly selected and tested. What are the probabilities associated with 0,1 , and 2 defective components being in the batch under each of the following conditions?

52. 某类型的组件以十件一批的方式运送给供应商。假设 $50\%$ 的批次中没有任何缺陷组件， $30\%$ 的批次中有一个缺陷组件，而 $20\%$ 的批次中有两个缺陷组件。从一批中随机选取两个组件进行测试。在以下每种条件下，批次中有0、1和2个缺陷组件的概率分别是多少？

53. Neither tested component is defective.

54. 两个测试组件都没有缺陷。

55. One of the two tested components is defective. [Hint: Draw a tree diagram with three first-generation branches for the three different types of batches.]

56. 两个测试组件中有一个是缺陷组件。 [提示：绘制一个树状图，有三个第一代分支，分别代表三种不同类型的批次。]

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57. Blue Cab operates $15\%$ of the taxis in a certain city, and Green Cab operates the other $85\%$ . After a nighttime hit-and-run accident involving a taxi, an eyewitness said the vehicle was blue. Suppose, though, that under night vision conditions, only $80\%$ of individuals can correctly distinguish between a blue and a green vehicle. What is the (posterior) probability that the taxi at fault was blue? In answering, be sure to indicate which probability rules you are using. [Hint: A tree diagram might help. Note: This is based on an actual incident.]

58. Blue Cab 在某个城市运营着 $15\%$ 的出租车，而 Green Cab 运营着其余的 $85\%$ 。在一次夜间肇事逃逸事故中，一名目击者称肇事车辆是蓝色的。然而，在夜间视觉条件下，只有 $80\%$ 的个人能够正确区分蓝色和绿色车辆。那么，肇事出租车是蓝色的（后验）概率是多少？在回答时，请务必说明您使用了哪些概率规则。[提示：树状图可能会有所帮助。注意：这是基于一个真实事件。]

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59. For customers purchasing a refrigerator at a certain appliance store, let $A$ be the event that the refrigerator was manufactured in the U.S., $B$ be the event that the refrigerator had an icemaker, and $C$ be the event that the customer purchased an extended warranty. Relevant probabilities are

60. 对于在某一电器商店购买冰箱的客户，设 $A$ 为冰箱是在美国制造的事件， $B$ 为冰箱带有制冰机的事件， $C$ 为客户购买了延长保修服务的事件。相关的概率如下：

$$P\left(A\right)=.75P\left(B\mid A\right)=.9P\left(B\mid A^{\prime }\right)=.8$$ $$P\left(C\mid A\cap B\right)=.8P\left(C\mid A\cap B^{\prime }\right)=.6$$ $$P\left(C\mid A^{\prime }\cap B\right)=.7P\left(C\mid A^{\prime }\cap B^{\prime }\right)=.3$$

1. Construct a tree diagram consisting of first-, second-, and third-generation branches, and place an event label and appropriate probability next to each branch.

2. 构建一个包含第一代、第二代和第三代分支的树状图，并在每个分支旁边放置一个事件标签和相应的概率。

3. Compute $P\left(A\cap B\cap C\right)$ .

4. 计算 $P\left(A\cap B\cap C\right)$ 。

5. Compute $P\left(B\cap C\right)$ .

6. 计算 $P\left(B\cap C\right)$ 。

7. Compute $P\left(C\right)$ .

8. 计算 $P\left(C\right)$ 。

9. Compute $P\left(A\mid B\cap C\right)$ , the probability of a U.S. purchase given that an icemaker and extended warranty are also purchased.

10. 计算 $P\left(A\mid B\cap C\right)$ ，即在购买制冰机和延长保修服务的情况下，购买美国制造冰箱的概率。

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11. The Reviews editor for a certain scientific journal decides whether the review for any particular book should be short (1-2 pages), medium (3-4 pages), or long (5-6 pages). Data on recent reviews indicates that $60\%$ of them are short, $30\%$ are medium, and the other $10\%$ are long. Reviews are submitted in either Word or LaTeX. For short reviews, $80\%$ are in Word, whereas $50\%$ of medium reviews are in Word and $30\%$ of long reviews are in Word. Suppose a recent review is randomly selected.

12. 某科学期刊的评论编辑决定任何特定书籍的评论应该是短的（1-2页）、中等的（3-4页）还是长的（5-6页）。最近评论的数据表明， $60\%$ 是短的， $30\%$ 是中等的，其余的 $10\%$ 是长的。评论可以提交为Word或LaTeX格式。对于短评论， $80\%$ 是Word格式的，而 $50\%$ 的中等评论和 $30\%$ 的长评论是Word格式的。假设最近随机选取了一篇评论。

13. What is the probability that the selected review was submitted in Word format?

14. 选中的评论是以Word格式提交的概率是多少？

15. If the selected review was submitted in Word format, what are the posterior probabilities of it being short, medium, or long?

16. 如果选中的评论是以Word格式提交的，那么它分别是短、中或长评论的后验概率是多少？

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17. A large operator of timeshare complexes requires anyone interested in making a purchase to first visit the site of interest. Historical data indicates that $20\%$ of all potential purchasers select a day visit, $50\%$ choose a one-night visit, and $30\%$ opt for a two-night visit. In addition, $10\%$ of day visitors ultimately make a purchase, $30\%$ of one-night visitors buy a unit, and $20\%$ of those visiting for two nights decide to buy. Suppose a visitor is randomly selected and is found to have made a purchase. How likely is it that this person made a day visit? A one-night visit? A two-night visit?

18. 一个大型分时度假村运营商要求有意购买的人首先参观感兴趣的地点。历史数据显示， $20\%$ 的潜在购买者选择日间访问， $50\%$ 选择一晚访问，而 $30\%$ 选择两晚访问。此外， $10\%$ 的日间访问者最终进行了购买， $30\%$ 的一晚访问者购买了单位，而 $20\%$ 的两晚访问者决定购买。假设随机选择了一位进行了购买的访客，这位访客进行日间访问的可能性有多大？一晚访问的可能性有多大？两晚访问的可能性有多大？

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19. Consider the following information about travelers on vacation (based partly on a recent Travelocity poll): $40\%$ check work email, $30\%$ use a cell phone to stay connected to work, $25\%$ bring a laptop with them, $23\%$ both check work email and use a cell phone to stay connected, and $51\%$ neither check work email nor use a cell phone to stay connected nor bring a laptop. In addition, 88 out of every 100 who bring a laptop also check work email, and 70 out of every 100 who use a cell phone to stay connected also bring a laptop.

20. 考虑以下关于度假旅客的信息（部分基于最近Travelocity的一项调查）： $40\%$ 检查工作邮件， $30\%$ 使用手机保持与工作的联系， $25\%$ 带笔记本电脑， $23\%$ 既检查工作邮件又使用手机保持与工作的联系， $51\%$ 既不检查工作邮件也不使用手机保持与工作的联系，也不带笔记本电脑。此外，每100个带笔记本电脑的人中有88个也检查工作邮件，而每100个使用手机保持联系的人中有70个也带笔记本电脑。

21. What is the probability that a randomly selected traveler who checks work email also uses a cell phone to stay connected?

22. 随机选择一个检查工作邮件的旅客，他/她同时使用手机保持联系的概率是多少？

23. What is the probability that someone who brings a laptop on vacation also uses a cell phone to stay connected?

24. 带笔记本电脑度假的人同时使用手机保持联系的概率是多少？

25. If the randomly selected traveler checked work email and brought a laptop, what is the probability that he/ she uses a cell phone to stay connected?

26. 如果随机选择的旅客检查了工作邮件并带了笔记本电脑，那么他/她使用手机保持联系的概率是多少？

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27. There has been a great deal of controversy over the last several years regarding what types of surveillance are appropriate to prevent terrorism. Suppose a particular surveillance system has a $99\%$ chance of correctly identifying a future terrorist and a $99.9\%$ chance of correctly identifying someone who is not a future terrorist. If there are 1000 future terrorists in a population of 300 million, and one of these 300 million is randomly selected, scrutinized by the system, and identified as a future terrorist, what is the probability that he/she actually is a future terrorist? Does the value of this probability make you uneasy about using the surveillance system? Explain.

28. 在过去几年里，关于预防恐怖主义应采取何种监视手段引发了大量争议。假设一个特定的监视系统有 $99\%$ 的概率正确识别出未来的恐怖分子，有 $99.9\%$ 的概率正确识别出不是未来恐怖分子的人。如果在一个拥有3亿人口中有1000个未来恐怖分子，随机选择这3000万人中的一人，通过系统审查并被识别为未来恐怖分子，那么他/她实际上真的是未来恐怖分子的概率是多少？这个概率的值是否让你对使用监视系统感到不安？请解释。

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29. A friend who lives in Los Angeles makes frequent consulting trips to Washington, D.C.; $50\%$ of the time she travels on airline #1, $30\%$ of the time on airline #2, and the remaining $20\%$ of the time on airline #3 . For airline #1, flights are late into D.C. $30\%$ of the time and late into L.A. $10\%$ of the time. For airline #2, these percentages are $25\%$ and $20\%$ , whereas for airline #3 the percentages are $40\%$ and $25\%$ . If we learn that on a particular trip she arrived late at exactly one of the two destinations, what are the posterior probabilities of having flown on airlines $\#1,\#2$ , and $\#3$ ? Assume that the chance of a late arrival in L.A. is unaffected by what happens on the flight to D.C. [Hint: From the tip of each first-generation branch on a tree diagram, draw three second-generation branches labeled, respectively, 0 late, 1 late, and 2 late.]

30. 我有一个住在洛杉矶的朋友，她经常往返于华盛顿特区进行咨询； $50\%$ 的时间她乘坐第一家航空公司， $30\%$ 的时间乘坐第二家航空公司，其余 $20\%$ 的时间乘坐第三家航空公司。对于第一家航空公司，飞往华盛顿特区的航班 $30\%$ 的时间晚点，飞往洛杉矶的航班 $10\%$ 的时间晚点。对于第二家航空公司，这些百分比分别是 $25\%$ 和 $20\%$ ，而对于第三家航空公司，这些百分比分别是 $40\%$ 和 $25\%$ 。如果我们得知在特定的一次旅行中，她恰好在一个目的地晚点到达，那么她乘坐第一家、第二家和第三家航空公司的后验概率分别是多少？假设在洛杉矶晚点到达的概率不受飞往华盛顿特区航班情况的影响。[提示：在每个第一代分支的顶端，画出三个分别标记为0次晚点、1次晚点和2次晚点的第二代分支。]

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31. In Exercise 59, consider the following additional information on credit card usage:

32. 在练习59中，考虑以下关于信用卡使用的额外信息：

$70\%$ of all regular fill-up customers use a credit card.

$70\%$ 的所有定期加油客户使用信用卡。

$50\%$ of all regular non-fill-up customers use a credit card.

所有常规非加满油的顾客 $50\%$ 使用信用卡。

$60\%$ of all plus fill-up customers use a credit card.

所有加满油的顾客 $60\%$ 使用信用卡。

$50\%$ of all plus non-fill-up customers use a credit card.

所有加满油非加满油的顾客 $50\%$ 使用信用卡。

$50\%$ of all premium fill-up customers use a credit card.

所有高级加满油的顾客 $50\%$ 使用信用卡。

$40\%$ of all premium non-fill-up customers use a credit card.

所有高级非加满油的顾客 $40\%$ 使用信用卡。

Compute the probability of each of the following events for the next customer to arrive (a tree diagram might help).

计算以下每个事件在下一个到达的顾客中发生的概率（树状图可能会有所帮助）。

1. {plus and fill-up and credit card }

2. {加满油且高级且使用信用卡 }

3. {premium and non-fill-up and credit card }

4. {高级且非加满油且使用信用卡 }

5. {premium and credit card }

6. {高级且使用信用卡 }

7. {fill-up and credit card }

8. {加满油且使用信用卡 }

9. {credit card }

10. {使用信用卡 }

11. If the next customer uses a credit card, what is the probability that premium was requested?

12. 如果下一个顾客使用信用卡，那么请求高级油的概率是多少？