Schaum's Outline of Elements of Statistics I: Descriptive Statistics and Probability

Stephen Bernstein, Ruth Bernstein

Chapter 9 PROBABILITY: CALCULATING RULES AND COUNTING RULES - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider again the experiment described in Example 9.1 in which a vaccine is tested on a group of 160 volunteers. Eighty volunteers are vaccinated and the rest are not. After 12 months all 160 people are asked if they got a cold during the past year. The results are shown in Table 9.1. The experiment is to randomly select one of the 160 people. If for this experiment $S=\{$ the 160 people\}, $A=\{$ got a cold $\}, A^{\prime}=\{$ did not get a cold $\}, B=\{$ vaccinated $\}$, and $B^{\prime}=\{$ not vaccinated $\}$, then find these probabilities:
(a) $P\left(A \mid B^{\prime}\right)$,
(b) $P(B \mid A)$,
(c) $P\left(B^{\prime} \mid A^{\prime}\right)$.

Harsh Gadhiya

Numerade Educator

02:53

Problem 2

If in Problem 9.1, two people are randomly selected, one after the other, from the 160, then what is the probability that:
(a) both got colds,
(b) one got a cold and the other did not?

Anand Jangid

Numerade Educator

02:33

Problem 3

In the work force of a large factory, $70 \%$ of the employees are high-school graduates, $8 \%$ are supervisors, and $5 \%$ are both supervisors and high-school graduates. From this information, answer these questions. (a) If the employee is a high-school graduate, what is the probability he is a supervisor? (b) If the employee is not a high-school graduate, what is the probability be is a supervisor?

Pratyush Raitan

Numerade Educator

01:25

Problem 4

A die is thrown two times. What is the probability that the total of both times is six, given that the first is twice as large as the second?

Prabhakar Kumar

Numerade Educator

03:14

Problem 5

Two cards are drawn from a well-shuffled, standard deck of playing cards [four suits (diamonds, hearts, clubs, and spades) of 13 cards each (ace through king)] If the first card is not replaced between selections, then what is the probability that: $(a)$ both cards will be hearts,
(b) both will be queens, (c) one will be a king and the other a queen?

Chai Santi

Numerade Educator

02:55

Problem 6

A manufacturer of automobile headlights has sent a shipment of 1,000 headlights to a customer, not knowing that three of the headlights are defective. The customer has a policy of testing a sample of three headlights from such a shipment, and if at least one of the three is defective he rejects the shipment. What is the probability he will reject this shipment?

Manisha Sarker

Numerade Educator

00:46

Problem 7

In Chapter 8, we used the set theory interpretation of probability to show for the experiment of flipping a coin twice that
$$
P(H H, H T, T H)=P(\text { at least one head })=0.75
$$

Now, use equation (9.8) (the special multiplication rule) to show that this is true.

Joshua Eastwood

Numerade Educator

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Problem 8

You know that $P(A)=0.25$ and $P(A \cap B)=0.20$. What is $P(B \mid A)$ if; $\quad(a) A$ and $B$ are independent events,
(b) $A$ and $B$ are dependent events?

Dani Dutmer

Numerade Educator

01:00

Problem 9

In the cold-vaccination study described in Example 9.1, 100 people got colds and 60 people did not get colds (see Table 9.1). Four consecutive random selections are made from the 160 people, with replacement after each selection. What is the probability that the first two had colds and the second two did not?

Varsha Aggarwal

Numerade Educator

00:45

Problem 10

You flip a coin seven times in a row and get seven heads. (a) What is the probability of this occurring? (b) What is the probability that if you go on to flip the coin an eighth time you will get a tail?

Lourence Gonhovi

Numerade Educator

01:37

Problem 11

What is the probability of rolling a 2 at least once in three consecutive rolls of a die?

Ashley High

Numerade Educator

00:44

Problem 12

For the cold-vaccination study in Example 9.1 (see Table 9.1), if $S=\{$ the 160 people \}, $A=\{$ got a cold\}, $A^{\prime}=$ \{did not get a cold \}, $B=\{$ was vaccinated $\}, B^{\prime}=$ \{was not vaccinated $\}$, then use equation (9.10) (the general addition rule) to determine:
(a) $P(A \cup B)$,
(b) $P\left(A^r \cup B\right)$,
(c) $P\left(A \cup B^{\prime}\right)$,
(d) $P\left(A^{\prime} \cup B^{\prime}\right)$,
(e) $P\left(A \cup A^{\prime}\right)$

Nick Johnson

Numerade Educator

03:42

Problem 13

In the Venn diagram in Fig. 9-6, the numbers on the boundaries of the circles are the probabilities for the events represented by the circles, and the numbers within enclosed areas of the circles are the probabilities for events represented by those areas. Using this information and equation (9.10), find:
(a) $P(A \cup B)$,
(b) $P\left(A^{\prime} \cup B\right)$,
(c) $P\left(A \cup B^{\prime}\right)$,
(d) $P\left(A^{\prime} \cup B^{\prime}\right)$.

Joshua Argo

Numerade Educator

01:50

Problem 14

For the sample space shown in Fig. 9-6, are $A$ and $B$ independent events? Are $A$ and $B$ mutually exclusive events?

Harsh Gadhiya

Numerade Educator

03:42

Problem 15

In the Venn diagram in Fig. 9-7, the numbers in the circles are the probabilities for the events represented by the circles. Using this information and the appropriate addition rule, find:
(a) $P(A \cup B)$,
(b) $P\left(A^{\prime} \cup B\right)$,
(c) $P\left(A \cup B^{\prime}\right)$.
(d) $P\left(A^{\prime} \cup B^{\prime}\right)$.

Joshua Argo

Numerade Educator

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Problem 16

For the sample space shown in Fig. 9-7, are $A$ and $B$ independent events?

Hoan Nguyen

Numerade Educator

00:47

Problem 17

In the die-rolling experiment, if $S=\{1,2,3,4,5,6\}, A=\{$ even number $\}, B=\{$ number $\geq 2\}$, $C=\{$ number $\leq 4\}$, use equation (9.11) to determine $P(A \cup B \cup C)$.

James Macpherson

Numerade Educator

02:39

Problem 18

An wrn is an opaque vessel whose contents cannot be seen. Three such ums $\left(A_1, A_2, A_3\right)$ each contain four balls. The balls are identical except for color: $A_1$ has three blue balls and one yellow, $A_2$ has two blue and two yellow, and $A_3$ has one blue and three yellow. The experiment is to randomly select one of the urns and then, without looking, select one ball from that um. Let $A_1, A_2, A_3$ represent the events of selecting a ball from the given um, $B$ represent the event of selecting a blue ball, and $B^{\prime}$ the event of selecting a yellow ball. From this information develop a joint probability table.

Ramon Kryzhan

Numerade Educator

01:17

Problem 19

From the information in Problem 9.18, answer the following questions both directly from Table 9.5 and also by using equation (9.15) (Bayes' theorem). (a) Given that a blue ball was selected, what is the probability it came from um $A_1$ ? (b) Given that a yellow ball was selected, what is the probability it came from um $A_2$ ?

Ivan Kochetkov

Numerade Educator

03:50

Problem 20

A district sales manager for a textbook publishing company feels there is a $60 \%$ probability that a rival company will sell its chemistry textbook to the ehemistry department of a large university. He also feels that if this happens, then there is an $80 \%$ probability that a community college in the same city as the university, which will choose a chemistry textbook after the university, will also adopt the rival's book. If the university does not adopt, then he feels there is still a $50 \%$ probability that the college will adopt the rival's book anyway. If $U$ and $U$ represent the events of adoption and nonadoption by the university of the rival's book and $C$ and $C$ represent these events for the college, then develop a joint probability table that includes these intersection probabilities: $P(C \cap U)$, $P\left(C \cap U^{\prime}\right), P(C \cap U), P\left(C \cap U^{\prime}\right)$; and these marginal probabilities: $P\left(U^{\prime}\right), P\left(U^{\prime}\right), P(C), P\left(C^{\prime}\right)$.

Maxime Rossetti

Numerade Educator

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Problem 21

From the information in Problem 9.20, answer the following questions both directly from Table 9.6 and by using equation (9.15). (a) Given that the college has adopted the textbook, what is the probability that the university has also adopted it? (b) Given that the college has not adopted the textbook, what is the probability that the university has also not adopted it?

Rashmi Sinha

Numerade Educator

03:33

Problem 22

A new viral disease has infected approximately $25 \%$ of the pig population on farms in several southern states. There is a diagnostic test for the presence of the virus but it gives a positive result (virus present) only $84 \%$ of the time when the pig actually has the disease and a negative result (virus absent) only $80 \%$ of the time when the pig does not have the disease. The probability experiment is to take a pig from this population and test it for the presence of the virus. From this information, answer these questions. (a) If the test result is positive, what is the probability the pig actually is infected by the virus? (b) If the test result is negative, what is the probability the pig is actually not infected by the virus?
(c) What is the probability the test will give the correct diagnosis?

Adam Dehollander

Numerade Educator

Problem 23

An insurance company executive has developed an aptitude test for selling insurance. She knows that in the current sales force, $65 \%$ of the salespeople have good sales records and the remaining $35 \%$ have bad sales records. She gives her test to the entire sales force and finds that $73 \%$ of those with good records pass the test and $78 \%$ of those with bad records fail the test. The probability experiment is to select a salesperson at random and give them the test. From this information, answer these questions. (a) If someone passes the test, what is the probability they have a good sales record? (b) If someone fails the test, what is the probability they have a bad sales record? (c) What is the probability that performance on the test will correctly identify someone with either a good or a bad sales record?

Check back soon!

08:50

Problem 24

Four balls in an urn are identical except for color; one is red $(R)$, one white $(W)$, one yellow $(Y)$, and the fourth is blue $(B)$. The experiment is to pick a ball from the urn and then, without replacing it, pick a second ball. Use a tree diagram to find $P$ (at least one $Y$ ball).

Oluwadamilola Ameobi

Numerade Educator

04:08

Problem 25

A friend in high school wants to go to medical school, but first she will either go to a local college where she has already been accepted, or to a prestigious university. She would prefer to go to the university but feels there is only a $65 \%$ probability she will be accepted. She further thinks that if she goes to the college there is a $95 \%$ probability she will graduate and then a $50 \%$ probability she will be accepted by a medical school. If instead she goes to the university, then she feels there is a $70 \%$ probability she will graduate followed by a $75 \%$ probability she will be accepted by a medical school. Of course she has to graduate from either the college or the university to be accepted by a medical school. Use a tree diagram to determine $P$ (acceptance by a medical school).

James Kiss

Numerade Educator

01:05

Problem 26

A senator campaigning for reelection is trying to raise $$\$ 100,000$$ for a last-minute series of television advertisements. He thinks there is a $55 \%$ probability he will be able to raise the money and that if he does there is then a $70 \%$ chance he will be reelected. He also feels that if he fails to raise the money then there will still be a $55 \%$ probability he will be reelected. Use a tree diagram to find $P$ (he did not raise the money, given he was reelected).

Lourence Gonhovi

Numerade Educator

02:00

Problem 27

Use the counting rule: multiplication princlple to determine the number of sample points in the sample spaces for these experiments: (a) flipping a coin eleven times, (b) selecting a card from a standard 52 -card deck five times in a row with replacement and reshuffling after each selection.

Joshua Hale

Numerade Educator

01:00

Problem 28

For the experiments in Problem 9.27, determine the following probabilities using both equation (9.9) and Property 7 from Section 8.6:
(a) $P$ (getting a head on all 11 trials),
(b) $P$ (getting a queen of hearts on all 5 trials).

Christopher Stanley

Numerade Educator

01:54

Problem 29

A car manufacturer offers several options for a particular model of car: (a) two or four doors, (b) one of six colors (red, yellow, blue, green, white, silver),
(c) AM or AM-FM radio,
(d) automatic or manual shifting. How many versions of this car are possible? If each of the versions is described on a separate card, all of the cards are put in a bowl, and then you blindly select one of the cards from the bowl, what is the probability it will read: a red car with two doors, manual shifting, and an AM-FM radio?

Derrick Hanson

Numerade Educator

01:28

Problem 30

Use both equations (9.17) and (9.18) to determine how many ways eight books can be arranged in a line along a shelf.

Kyler Gray

Numerade Educator

01:13

Problem 31

After shuffling a standard deck of 52 cards you deal three cards which you place on a table in a leftto-right sequence. What is the probability that the sequence is jack, queen, king of the same suit?

Wendi Zhao

Numerade Educator

00:46

Problem 32

You live in a state where the car license plates have three letters (without duplications on a plate) on the left followed by three numbers (again without duplications) on the right. The letters are randomly selected from the 26 letters of the alphabet, and the numbers are randomly selected from the ten integers 0 to 9. Assuming that all plates are available, and that you are assigned a plate randomly, what is the probability that you will get a plate that reads: ABC012?

Maxime Rossetti

Numerade Educator

01:01

Problem 33

In the house of representatives of your state legislature there are 90 Democrats and 70 Republicans. By a random process, a Majority Leader and an Assistant Majority Leader will be selected from the Democrats and a Minority Leader and Assistant Minority Leader will be selected from the Republicans. How many permutations are there of these four leadership positions?

Kyler Gray

Numerade Educator

02:21

Problem 34

If in the house of representatives described in Problem 9.33 there are six Democrats and four Republicans from the same city, what is the probability that representatives from this city will be selected to all four leadership positions?

Lourence Gonhovi

Numerade Educator

00:54

Problem 35

If all three-letter words that can be formed from the word CLOVER (with no letter duplications in a word) are each written on a separate card and all these cards are placed in a bowl, what is the probability of then selecting a card from the bowl that has a word on it that begins with a vowel?

Joshua Sieverding

Numerade Educator

02:45

Problem 36

A manufacturer shows 40 bathing suits to a buyer. Use equation (9.20) to determine how many ways the buyer can choose five of the suits to sell in her store.

Lauren Shelton

Numerade Educator

01:56

Problem 37

Your instructor in a college history course gives you a list of 20 possible essay questions from which he will randomly pick four for the final examination. Pressed for time, you prepare for only four of the questions. What is the probability these four will be on the examination?

Steven Clarke

Numerade Educator

01:44

Problem 38

To play your state lottery, you select six numbers from 1 to 42. You win the grand prize if your six match the six winning numbers selected by the lottery. What is the probability that your ticket will win the grand prize?

Lourence Gonhovi

Numerade Educator

01:11

Problem 39

In the lottery in Problem 9.38, you win the second prize if you match five of the lottery's six winning numbers. What is the probability that your ticket will win the second prize?

Joshua Eastwood

Numerade Educator

01:29

Problem 40

What is the probability that if you are dealt a five-card poker hand, two of the cards will be diamonds, two will be hearts, and one will be a club?

James Chok

Numerade Educator

01:36

Problem 41

You want to arrange eight books in a line along a shelf. How many unique combinations are there of these eight books?

Linh Vu

Numerade Educator

01:01

Problem 42

A single card is drawn from a well-shuffled, standard deck of playing eards. Given that a diamond card has been selected, what is the probability it is a face card (jack, queen, or king)?

Charles Carter

Numerade Educator

01:45

Problem 43

A single card is drawn from a well-shuffled, standard deck of playing cards. Given that a card numbered 2,3, 4 , or 5 has been selected, what is the probabelity it is a diamond?

J Hardin

Numerade Educator

02:00

Problem 44

A single card is drawn from a well-shuffled, standard deck of playing eards. Given that a king has been selected, what is the probability it is a red card?

Raj Bala

Numerade Educator

01:12

Problem 45

Two cards are drawn, one after the other, from a well-shuffed, standard deck of playing cards. The first card is not returned to the deck after it has been drawn. Use equation (9.4) to determine the probability of selecting a diamond eard in both draws.

J Hardin

Numerade Educator

01:00

Problem 46

An urm contains 60 marbles: 30 are white, 18 are red, and 12 are blue. Two marbles are removed, at random, from the urn. The first marble is not retumed to the urn after it has been removed. What is the probability that both marbles are blue?

Wendi Zhao

Numerade Educator

02:49

Problem 47

Determine the probability of not rolling a 4 on either of two consecutive repetitions of the dierrolling experiment.

Dalia Rodriguez

Numerade Educator

01:12

Problem 48

In three repetitions of the card-selection experiment, if the cards are not replaced between selections, then what is the probabelity that they will all be queens?

Harsh Gadhiya

Numerade Educator

00:44

Problem 49

In the coin-flipping experiment, what is the probability of rolling two heads in succession?

Sheryl Ezze

Numerade Educator

04:19

Problem 50

In six repetitions of the card-selection experiment, with replacement and reshufling after each selection, what is the probability that all six cards will be red cards?

Ahmad Reda

Numerade Educator

00:39

Problem 51

During the past year in a maternity ward of a hospital, 1,060 males were bom and 1,000 females. Assuming these totals to be representative of all births, what is the probability that the next four babies bom in the ward will be girls?

Aditya Sood

Numerade Educator

02:00

Problem 52

A single card is drawn from a well-shuffled, standard deck of playing cards. What is the probability of drawing ether a diamond or a red card?

Raj Bala

Numerade Educator

00:36

Problem 53

A single card is drawn from a well-shuffled, standard deck of playing cards. What is the probability of drawing either a diamond or a king?

Sheryl Ezze

Numerade Educator

01:00

Problem 54

A single card is drawn from a well-shuffled, standard deck of playing cards. What is the probability of drawing either a 2,3 , or 4 and not a diamond?

Lynn Larson

Numerade Educator

02:09

Problem 55

Deternine $P(B \mid A)$ when:
(a) $A$ and $B$ are mutually exclusive events,
(b) $A$ and $B$ are independent events.
Ans. (a) 0 ,
(b) $P(B)$

Sanchit Jain

Numerade Educator

02:04

Problem 56

Determine $P(A \cap B)$ when:
(a) $A$ and $B$ are mutually exclusive events,
(b) $A$ and $B$ are independent events.
Ans. $(A) 0$,
(b) $P(A) P(B)$

Lucas Finney

Numerade Educator

02:19

Problem 57

Determine $P(A \cup B)$ when:
(a) $A$ and $B$ are mutually exclusive events;
(b) $A$ and $B$ are independent events.

Lucas Finney

Numerade Educator

02:07

Problem 58

One hundred mayors of U.S. cities are attending a conference on environmental issues Fifty of the mayors are Democrats and 50 are Republicans. Sixty are men and 40 are women, and of the 60 men, 25 are Democrats. If one of the mayors is randomly chosen, then deternine the probability that: (a) the mayor will be a male Democrat, (b) the mayor will be either a male or a Democrat.

Christopher Stanley

Numerade Educator

02:07

Problem 59

For the mayors in Problem 9.58, 15 of the 40 women are Republicans. If cene mayor is randomly selected from all 100 mayors each day for two days (sampling with replacement), then detemine the probability that (a) a man will be chosen on day one and a woman on day two, (b) on day two, either a male Democrat or a female Republican will be chosen.

Christopher Stanley

Numerade Educator

01:19

Problem 60

For the mayors in Problem 9.58, 15 of the 25 make Democrats are over 45 years of age. If one mayor is randomly selected from the 100, what is the probability of selecting a male Democrat who is over 45 years of age?

Jon Southam

Numerade Educator

Problem 61

For the mayors in Problem 9.58, 50 are over 45 years of age ( 30 males and 25 females). Of these 50, 25 are Democrats. If one mayor is randomly selected from the 100 , what is the probability of selecting either a male or a Democrat or sceneone who is over 45 years of age?

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01:45

Problem 62

Events $A_1, A_2$, and $A_3$ are mutually exclusive and exhaustive, with probabilities $P\left(A_1\right)=0.20, P\left(A_2\right)=0.60$, and $P\left(A_3\right)=0.20$. Given that $P\left(B \mid A_1\right)=0.10, P\left(B \mid A_2\right)=0.50$, and $P\left(B \mid A_3\right)=0.40$, calculate $P(B)$.

Haggai Liu

Numerade Educator

Problem 63

From the information provided in Problem 9.62, calculate:
(a) $P\left(A_1 \mid B\right)$,
(b) $P\left(A_2 \mid B\right)$,
(c) $P\left(A_3 \mid B\right)$

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04:57

Problem 64

Events $D_1, D_2$, and $D_3$ are mutually exclusive and exhaustive causes of two effects, $C$ and $C$. The probabilities of the causes are $P\left(D_1\right)=0.69, P\left(D_2\right)=0.05$, and $P\left(D_3\right)=0.26$. Given that $P\left(C \mid D_1\right)=0.10$, $P\left(C D_2\right)=0.36$, and $P\left(C D_3\right)=0.54$, what is the probability of effect $C^{\prime}$ ?

Anas Venkitta

Numerade Educator

01:44

Problem 65

For the information provided in Problem 9.64, what is the probability that effect $C^{\prime}$ is caused by $D_1, D_2$, or $D_3$ ? In other words, what are:
(a) $P\left(D_1 \mid C^{\prime}\right)$,
(b) $P\left(D_2 \mid C^n\right)$.
(c) $P\left(D_3 \mid C^{\prime}\right)$ ?

Aman Gupta

Numerade Educator

00:29

Problem 66

At Easter, a father places colored eggs in baskets for his small daughter to find. He hides three baskets in three hiding places. The first basket contains two red eggs, the second basket contains a red egg and a blue egg, and the third basket contains two blue eggs Given that the child finds a blue egg, what is the probability that it comes from the second basket?

Kerry Thornton-Genova

Numerade Educator

04:45

Problem 67

Colored marbles are placed in two urns, one black and the other white. The black urn contains 12 blae marbles and 6 red ones. The white um contains 4 blue marbles and 8 red ones. An um is selected at random and one marble is drawn from it If the marble is blue, then what is the probability that the marble was drawn from the black um?

Sonam Khatri

Numerade Educator

02:20

Problem 68

Foe the experiment of flipping a coin three times, use a tree diagram to find $P$ (at least two tails). Let $H_1, H_2$. and $\mathrm{H}_3$ represent besds on the first, second, and third llips, and $T_1, T_2$, and $T_3$ represent getting a tail on the first, second, and third flips.

Abdul Vahid M

Numerade Educator

03:08

Problem 69

For the experiment of rolling a die twice, use a tree diagram to find P(total for both rolk of 7 or 11 ). Let $1 a$, $2 a, 3 a, 4 a, 5 a$, and $6 a$ represent the possible outcomes for the first roll and $1 b, 2 b, 3 b, 4 b, 5 b$, and $6 b$ represent the outcomes for the second roll.

Robin Corrigan

Numerade Educator

00:37

Problem 70

A committee consisting of one man and one woman is to be chosen from a group of 8 men and 15 women. How many possible coenmittees can be chosen?

Sneha Ravi

Numerade Educator

03:03

Problem 71

A fraternity plans to send four students-a freshenan, a sophomore, a junior, and a senior $\rightarrow$ a nationa] meeting. The volunteers for this group include four freshmen, four sophomores, eight janiors, and three seniors. How many different sets of four students, one from each class, are possible?

Alice .

Numerade Educator

01:10

Problem 72

A man packs four slacks, six shirts, and three ties for a trip. How many different "outfits" of these three kinds of clothing can he form?

Narayan Hari

Numerade Educator