11. Drawing a Card Suppose a single card is selected from a standard 52-card deck. What is the probability that the card drawn is a club? Now suppose a single card is drawn from a standard 52-card deck, but we are told that the card is black.
What is the probability that the card drawn is a club?
12. Drawing a Card Suppose a single card is selected from a standard 52-card deck. What is the probability that the card drawn is a king? Now suppose a single card is drawn from a standard 52-card deck, but we are told that the card is a heart. What is the probability that the card drawn is a king? Did the knowledge that the card is a heart change the probability that the card was a king? What is the term used to describe this result?1/13; 1/13; no; independence 13. Rainy Days For the month of June in the city of Chicago,
37% of the days are cloudy. Also in the month of June in the city of Chicago, 21% of the days are cloudy and rainy.
What is the probability that a randomly selected day in June will be rainy if it is cloudy? 0.568
14. Cause of Death According to the U.S. National Center for Health Statistics, in 2002, 0.2% of deaths in the United States were 25- to 34-year-olds whose cause of death was cancer. In addition, 1.97% of all those who died were 25 to 34 years old. What is the probability that a randomly se-lected death is the result of cancer if the individual is known to have been 25 to 34 years old?0.102
15. High School Dropout According to the U.S. Census Bureau, 9.1% of high school dropouts are 16- to 17-year-olds. In addi-tion, 5.8% of white high school dropouts are 16- to 17-year-olds.What is the probability that a randomly selected dropout is white, given that he or she is 16 to 17 years old?0.637 16. Income by Region According to the U.S Census Bureau,
19.1% of U.S. households are in the Northeast. In addi-tion, 4.4% of U.S. households earn $75,000 per year or more and are located in the Northeast. Determine the probability that a randomly selected U.S. household earns more than $75,000 per year, given that the household is lo-cated in the Northeast.0.230
17. Health Insurance Coverage. The following data repre-sent, in thousands, the type of health insurance coverage of people by age in the year 2002.
NW
NW
Applying the Concepts
18. Cigar Smoking. The data in the following table show the results of a national study of U.S. men that investi-gated the association between cigar smoking and death from cancer. Note: Current cigar smoker means “cigar smoker at time of death.”
137, 243
18–44 45–64
Private 49,473 76,294 52,520 20,685
Government 19,662 11,922 9,227 32,813
None 8,531 25,678 9,106 258
Source: U.S. Census Bureau
>64
<18
Age
Never smoked cigars 782 120,747
Former cigar smoker 91 7,757
Current cigar smoker 141 7,725
Source: Shapiro, Jacobs, and Thun. “Cigar Smoking in Men and Risk of Death from Tobacco-Related Cancers,” Journal of the National Cancer Institute, February 16, 2000
Died from Did Not Die Cancer from Cancer
Under 16 228 108
16–20 5696 2386
21–34 13,553 4148
35–54 14,395 5017
55–69 4937 1708
70 and over 3159 1529
Source: Traffic Safety Facts 2002. Federal Highway Ad-ministration, 2002
Age Male Female
1 4; 1
2
(a) What is the probability that a randomly selected indi-vidual from the study who died from cancer was a for-mer cigar smoker?0.090
(b) What is the probability that a randomly selected indi-vidual from the study who was a former cigar smoker died from cancer?0.012
19. Driver Fatalities The following data represent the num-ber of driver fatalities in the United States in 2002 by age for male and female drivers:
(a) What is the probability that a randomly selected driv-er fatality who was male was 16 to 20 years old?0.136 (b) What is the probability that a randomly selected
driv-er fatality who was 16 to 20 was male?0.705
(c) Suppose you are a police officer called to the scene of a traffic accident with a fatality. The dispatcher states that the victim is 16 to 20 years old, but the gender is not known. Is the victim more likely to be male or fe-male? Why?Male
(a) What is the probability that a randomly selected indi-vidual who is less than 18 years old has no health in-surance?0.110
(b) What is the probability that a randomly selected indi-vidual who has no health insurance is less than 18 years old?0.196
21. Acceptance Sampling Suppose you just received a ship-ment of six televisions. Two of the televisions are defec-tive. If two televisions are randomly selected, compute the probability that both televisions work. What is the proba-bility at least one does not work?0.4; 0.6
22. Committee A committee consists of four women and three men. The committee will randomly select two peo-ple to attend a conference in Hawaii. Find the probability that both are women.
23. Board Work This past semester, I had a small business calculus section. The students in the class were Mike, Neta, Jinita, Kristin, and Dave. Suppose I randomly select two people to go to the board to work problems. What is the probability that Dave is the first person chosen to go to the board and Neta is the second?
24. Party My wife has organized a monthly neighborhood party. Five people are involved in the group: Yolanda (my wife), Lorrie, Laura, Kim, and Anne Marie. They decide to randomly select the first and second home that will host the party. What is the probability that my wife hosts the first party and Lorrie hosts the second? Note: Once a home has hosted, it cannot host again until all other homes have hosted.
25. Playing a CD on the Random Setting Suppose a compact disk (CD) you just purchased has 13 tracks. After listening to the CD, you decide that you like 5 of the songs. With the random feature on your CD player, each of the 13 songs is played once in random order. Find the probability that among the first two songs played
(a) You like both of them. Would this be unusual?
(b) You like neither of them.
(c) You like exactly one of them.
(d) Redo (a)–(c) if a song can be replayed before all 13 songs are played (if, for example, track 2 can play twice in a row).
26. Packaging Error Due to a manufacturing error, three cans of regular soda were accidentally filled with diet soda and placed into a 12-pack. Suppose that two cans are ran-domly selected from the case.
(a) Determine the probability that both contain diet soda.
(b) Determine the probability that both contain regular soda. Would this be unusual?
(c) Determine the probability that exactly one is diet and one is regular.
27. Planting Tulips A bag of 30 tulip bulbs purchased from a nursery contains 12 red tulip bulbs, 10 yellow tulip bulbs, and 8 purple tulip bulbs.
(a) What is the probability that two randomly selected tulip bulbs are both red?0.152
(b) What is the probability that the first bulb selected is red and the second yellow?0.138
(c) What is the probability that the first bulb selected is yellow and the second is red?0.138
(d) What is the probability that one bulb is red and the other yellow?0.276
28. Golf Balls The local golf store sells an “onion bag” that contains 35 “experienced” golf balls. Suppose the bag con-tains 20 Titleists, 8 Maxflis, and 7 Top-Flites.
(a) What is the probability that two randomly selected golf balls are both Titleists?0.319
(b) What is the probability that the first ball selected is a Titleist and the second is a Maxfli?0.134
(c) What is the probability that the first ball selected is a Maxfli and the second is a Titleist?0.134
(d) What is the probability that one golf ball is a Titleist and the other is a Maxfli?0.269
29. Smokers According to the National Center for Health Statistics, there is a 23.4% probability that a randomly se-lected resident of the United States aged 25 years or older is a smoker. In addition, there is a 21.7% probability that a randomly selected resident of the United States aged 25 years or older is female, given that he or she smokes. What is the probability that a randomly selected resident of the United States aged 25 years or older is female and smokes? Would it be unusual to randomly select a resi-dent of the United States aged 25 years or older who is fe-male and smokes?0.051; no
Section 5.4 Conditional Probability and the General Multiplication Rule 263
20. Marital Status The following data, in thousands, represent the marital status of Americans 25 years old or older and their level of education in 2003.
Never married 4,333 8,637 7,403 8,321
Married, spouse present 14,787 35,376 28,892 34,693
Married, spouse absent 2,032 2,534 1,633 1,190
Separated 1,134 1,596 1,103 614
Widowed 4,582 5,155 2,487 1,746
Divorced 2,887 7,612 6,393 4,490
Source: Educational Attainment in the United States: 2003. U.S. Census Bureau, June 2004 Did Not
Graduate from High School College
High School Graduate Some College Graduate
(a) What is the probability that a randomly selected individual who has never married is a high school graduate?0.301
(b) What is the probability that a randomly selected individual who is a high school graduate has never married?0.142
25
35. Independence in Small Samples from Large Populations Suppose a computer chip company has just shipped 10,000 computer chips to a computer company. Unfor-tunately, 50 of the chips are defective.
(a) Compute the probability that two randomly select-ed chips are defective using conditional probability.
(b) There are 50 defective chips out of 10,000 shipped.
The probability that the first chip randomly selected is
defective is Compute the
probability that two randomly selected chips are de-fective under the assumption of independent events.
Compare your results to part (a). Conclude that, when small samples are taken from large populations with-out replacement, the assumption of independence does not significantly affect the probability.0.000025
50
10,000 = 0.005 = 0.5%.
NW 30. Multiple Jobs According to the U.S. Bureau of Labor
Statistics, there is a 5.84% probability that a randomly selected employed individual has more than one job (a multiple-job holder). Also, there is a 52.6% probability that a randomly selected employed individual is male, given that he has more than one job. What is the proba-bility that a randomly selected employed individual is a multiple-job holder and male? Would it be unusual to randomly select an employed individual who is a multi-ple-job holder and male?0.031; yes
(35a) 0.0000245 (34a) 0.00000154
31. The Birthday Problem Determine the probability that at least 2 people in a room of 10 people share the same birth-day, ignoring leap years and assuming each birthday is equally likely by answering the following questions:
(a) Compute the probability that 10 people have differ-ent birthdays. (Hint: The first person’s birthday can occur 365 ways; the second person’s birthday can occur 364 ways, because he or she cannot have the same birthday as the first person; the third person’s birthday can occur 363 ways, because he or she can-not have the same birthday as the first or second person; and so on.)0.883
(b) The complement of “10 people have different birth-days” is “at least 2 share a birthday.” Use this infor-mation to compute the probability that at least 2 people out of 10 share the same birthday.0.117
32. The Birthday Problem Using the procedure given in Problem 31, compute the probability that at least 2 peo-ple in a room of 23 peopeo-ple share the same birthday.
0.507
33. A Flush A flush in the card game of poker occurs if a player gets five cards that are all the same suit (clubs, di-amonds, hearts, or spades). Answer the following ques-tions to obtain the probability of being dealt a flush in five cards.
(a) We initially concentrate on one suit, say clubs. There are 13 clubs in a deck. Compute
P(first card is clubs and second card is clubs and third card is clubs and fourth card is clubs and fifth card is clubs).0.000495
(b) A flush can occur if we get five clubs or five dia-monds or five hearts or five spades. Compute P(five clubs or five diamonds or five hearts or five spades). Note the events are mutually exclusive.
0.002
P1five clubs2 =
34. A Royal Flush A royal flush in the game of poker occurs if the player gets the cards Ten, Jack, Queen, King, and Ace all in the same suit. Use the results of Problem 33 to compute the probability of being dealt a royal flush.
36. Independence in Small Samples from Large Popula-tions Suppose a poll is being conducted in the village of Lemont. The pollster identifies her target population as all residents of Lemont 18 years old or older. This pop-ulation has 6494 people.
(a) Compute the probability that the first resident select-ed to participate in the poll is Roger Cummings and the second is Rick Whittingham.0.0000000237 (b) The probability that any particular resident of
Lemont is the first person picked is Compute the probability that Roger is selected first and Rick is selected second, assuming independence. Com-pare your results to part (a). Conclude that, when small samples are taken from large populations without replacement, the assumption of independ-ence does not significantly affect the probability.
0.0000000237
1 6494.
37. Independent? Refer to the contingency table in Prob-lem 17 that relates age and health insurance coverage.
Determine P( years old) and
Are the events “ years old” and
“no health insurance” independent?No
38. Independent? Refer to the contingency table in Prob-lem 18 that relates cigar smoking and deaths from can-cer. Determine P(died from cancer) and
Are the events “died from cancer” and “current cigar smoker” independent?
No
39. Independent? Refer to the contingency table in Prob-lem 19 that relates age of driving fatality to gender.
De-termine P(female) and Are the
events “female” and “16–20” independent?No
40. Independent? Refer to the contingency table in Problem 20 that relates marital status and level of education. De-termine P(divorced) and
Are the events “divorced” and “college graduate” inde-pendent?No
P1divorced ƒ college graduate2.
P1female ƒ 16–202.
cigar smoker2.
from cancer ƒ current
P1died 618
health insurance62.18 P1618 years old ƒ no