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Excursions in Modern Mathematics

Peter Tannenbaum

Chapter 16

Probabilities, Odds, and Expectations - all with Video Answers

Educators

Chapter Questions

02:57

Problem 1

Using set notation, write out the sample space for each of the following random experiments.
(a) A coin is tossed three times in a row. The observation is how the coin lands ( $H$ or $T$ ) on each toss.
(b) A basketball player shoots three consecutive free throws. The observation is the result of each free throw $(s$ for success, $f$ for failure).
(c) A coin is tossed three times in a row. The observation is the number of times the coin comes up tails.
(d) A basketball player shoots three consecutive free throws. The observation is the number of successes.

Sneha Ravi
Sneha Ravi
Numerade Educator
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Problem 2

Using set notation, write out the sample space for each of the following random experiments.
(a) A coin is tossed four times in a row. The observation is how the coin lands ( $H$ or $T$ ) on each toss.
(b) A student randomly guesses the answers to a fourquestion true-or-false quiz. The observation is the student's answer ( $T$ or $F$ ) for each question.
(c) A coin is tossed four times in a row. The observation is the percentage of tosses that are heads.
(d) A student randomly guesses the answers to a fourquestion true-or-false quiz. The observation is the percentage of correct answers in the test.

James Kiss
James Kiss
Numerade Educator
02:57

Problem 3

Using set notation, write out the sample space for each of the following random experiments:
(a) Roll three dice. The observation is the total of the three numbers rolled.
(b) Toss a coin five times. The observation is the difference (# of heads-# of tails) in the five tosses.

Sneha Ravi
Sneha Ravi
Numerade Educator
01:28

Problem 4

Using set notation, write out the sample space for each of the following random experiments:
(a) Three runners $(A, B,$ and $C$ ) are running in a race (assume that there are no ties). The observation is the order in which the three runners cross the finish line.
(b) Four runners $(A, B, C,$ and $D)$ are running in a qualifying race (assume that there are no ties). The top two finishers qualify for the finals. The observation is the pair of runners that qualify for the finals.

Vysakh M
Vysakh M
Numerade Educator
01:47

Problem 5

The board of directors of Fibber Corporation has five members $(A, B, C, D,$ and $E)$. Using set notation write out the sample space for each of the following random experiments:
(a) A chairman and a treasurer are elected.
(b) Three directors are selected to form a search committee to hire a new CEO.

AG
Ankit Gupta
Numerade Educator
01:21

Problem 6

You reach into a large jar containing jelly beans of four different flavors [Juicy Pear $(J),$ Kiwi $(K),$ Licorice $(L),$ and Mango ( $M$ ) ] and grab two jelly beans at random. The observation is the flavors of the two jelly beans. Using set notation, write out the sample space for this experiment.

Madysn Cardinal
Madysn Cardinal
Numerade Educator
01:08

Problem 7

The sample spaces are too big to write down in full. In these exercises, you should describe the sample space either by describing a generic outcome or by listing some outcomes and then using the ... notation. In the latter case, you should write down enough outcomes to make the description reasonably clear.
A coin is tossed 10 times in a row. The observation is how the coin lands ( $H$ or $T$ ) on each toss. Describe the sample space.

Amy Jiang
Amy Jiang
Numerade Educator
03:28

Problem 8

The sample spaces are too big to write down in full. In these exercises, you should describe the sample space either by describing a generic outcome or by listing some outcomes and then using the ... notation. In the latter case, you should write down enough outcomes to make the description reasonably clear.
A student randomly guesses the answers to a 10-question true-or-false quiz. The observation is the student's answer ( $T$ or $F$ ) for each question. Describe the sample space.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
01:08

Problem 9

The sample spaces are too big to write down in full. In these exercises, you should describe the sample space either by describing a generic outcome or by listing some outcomes and then using the ... notation. In the latter case, you should write down enough outcomes to make the description reasonably clear.
A die is rolled four times in a row. The observation is the number that comes up on each roll. Describe the sample space.

Amy Jiang
Amy Jiang
Numerade Educator
03:28

Problem 10

The sample spaces are too big to write down in full. In these exercises, you should describe the sample space either by describing a generic outcome or by listing some outcomes and then using the ... notation. In the latter case, you should write down enough outcomes to make the description reasonably clear.
A student randomly guesses the answers to a multiplechoice quiz consisting of 10 questions. The observation is the student's answer $(A, B, C, D,$ or $E)$ for each question. Describe the sample space.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
03:34

Problem 11

A coin is tossed three times in a row. The observation is how the coin lands (heads or tails) on each toss [see Exercise $1(a)]$. Write out the event described by each of the following statements as a set.
(a) $E_{1}$ : "the coin comes up heads exactly twice."
(b) $E_{2}:$ "all three tosses come up the same."
(c) $E_{3}:$ "exactly half of the tosses come up heads."
(d) $E_{4}:$ "the first two tosses come up tails."

Bryan Meares
Bryan Meares
Numerade Educator
01:39

Problem 12

A student randomly guesses the answers to a four-question true-or-false quiz. The observation is the student's answer (T or $F$ ) for each question [see Exercise $2(\mathrm{~b})$ ]. Write out the event described by each of the following statements as a set.
(a) $E_{1}:$ "the student answers $T$ to two out of the four questions."
(b) $E_{2}$ : "the student answers $T$ to at least two out of the four questions."
(c) $E_{3}$ : "the student answers $T$ to at most two out of the four questions."
(d) $E_{4}:$ "the student answers $T$ to the first two questions."

TB
Timothy Behan
Numerade Educator
01:00

Problem 13

A pair of dice is rolled. The observation is the number that comes up on each die (see Example 16.4 ). Write out the event described by each of the following statements as a set.
(a) $E_{1}$ " pairs are rolled." (A "pair" is a roll in which both dice come up the same number.)
(b) $E_{2}:$ "craps are rolled." ("Craps" is a roll in which the sum of the two numbers rolled is $2,3,$ or $12 .)$
(c) $E_{3}:$ "a natural is rolled." (A "natural" is a roll in which the sum of the two numbers rolled is 7 or $11 .$ )

Ian Musser
Ian Musser
Numerade Educator
02:27

Problem 14

A card is drawn out of a standard deck of 52 cards. Each card can be described by giving its "value" $(A, 2,3,4, \ldots,$ $10, J, Q, K)$ and its "suit" ( $H$ for hearts, $C$ for clubs, $D$ for diamonds, and $S$ for spades). For example, $2 D$ denotes the two of diamonds and $J H$ denotes the jack of hearts. Write out the event described by cach of the following statements as a set.
(a) $E_{1}:$ "draw a queen."
(b) $E_{2}:$ "draw a heart."
(c) $E_{3}:$ "draw a face card." (A "face" card is a jack, queen, or king.)

Nick Johnson
Nick Johnson
Numerade Educator
02:27

Problem 14

A card is drawn out of a standard deck of 52 cards. Each card can be described by giving its "value" $(A, 2,3,4, \ldots,$ $10, J, Q, K)$ and its "suit" ( $H$ for hearts, $C$ for clubs, $D$ for diamonds, and $S$ for spades). For example, $2 D$ denotes the two of diamonds and $J H$ denotes the jack of hearts. Write out the event described by each of the following statements as a set.
(a) $E_{1^{*}}$ "draw a queen."
(b) $E_{2}:{ }^{*}$ draw a heart."
(c) $E_{3}$ " "draw a face card." (A "face" card is a jack, queen, or king.)

Nick Johnson
Nick Johnson
Numerade Educator
00:38

Problem 15

A coin is tossed 10 times in a row. The observation is how the coin lands ( $H$ or $T$ ) on each toss (see Exercise 7 ). Write out the event described by each of the following statements as a set.
(a) $E_{1}:$ "none of the tosses comes up tails."
(b) $E_{2}:$ "exactly one of the 10 tosses comes up tails"
(c) $E_{3}$ : "nine or more of the tosses come up tails."

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
01:26

Problem 16

Five candidates $(A, B, C, D,$ and $E)$ have a chance to be selected to be on American Idol. Any subset of them (including none of them or all of them) can be selected. The observation is which subset of individuals is selected. Write out the event described by each of the following statements as a set.
(a) $E_{1}:$ "two candidates get selected."
(b) $E_{2}:$ "three candidates get selected."
(c) $E_{3}:$ "three candidates get selected, and $A$ is not one of them."

Angela Guo
Angela Guo
Numerade Educator
02:21

Problem 17

A California license plate starts with a digit other than 0 followed by three capital letters followed by three more digits (0 through 9).
(a) How many different California license plates are possible?
(b) How many different California license plates start with $a 5$ and end with a $9 ?$
(c) How many different California license plates have no repeated symbols (all the digits are different and all the letters are different)?

Gregory Higby
Gregory Higby
Numerade Educator
02:18

Problem 18

A computer password consists of four letters ( $A$ through $Z$ ) followed by a single digit ( 0 through 9 ). Assume that the passwords are not case sensitive (i.e., that an uppercase letter is the same as a lowercase letter).
(a) How many different passwords are possible?
(b) How many different passwords end in $1 ?$
(c) How many different passwords do not start with $Z$ ?
(d) How many different passwords have no $Z$ 's in them?

Pratyush Raitan
Pratyush Raitan
Numerade Educator
00:31

Problem 19

Jack packs two pairs of shoes, one pair of boots, three pairs of jeans, four pairs of dress pants, and three dress shirts for a trip.
(a) How many different outfits can Jack make with these items?
(b) If Jack were also to bring along two jackets so that he could wear either a dress shirt or a dress shirt plus a jacket, how many outfits could Jack make?

Heather Zimmers
Heather Zimmers
Numerade Educator
01:06

Problem 20

A French restaurant offers a menu consisting of three different appetizers, two different soups, four different salads, nine different main courses, and five different desserts.
(a) A fixed-price lunch meal consists of a choice of appetizer, salad, and main course. How many different fixed-price lunch meals are possible?
(b) A fixed-price dinner meal consists of a choice of appetizer, a choice of soup or salad, a main course, and a dessert. How many different fixed-price dinner meals are possible?
(c) A dinner special consists of a choice of soup, salad, or both, plus a main course. How many dinner specials are possible?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:28

Problem 21

A set of reference books consists of eight volumes numbered 1 through 8 .
(a) In how many ways can the eight books be arranged on a shelf?
(b) In how many ways can the eight books be arranged on a shelf so that at least one book is out of order?

Kyler Gray
Kyler Gray
Numerade Educator
01:05

Problem 22

Nine people (four men and five women) line up at a checkout stand in a grocery store.
(a) In how many ways can they line up?
(b) In how many ways can they line up if the first person in line must be a woman?

AG
Ankit Gupta
Numerade Educator
01:05

Problem 23

Nine people (four men and five women) line up at a checkout stand in a grocery store.
(a) In how many ways can they line up if all five women must be at the front of the line?
(b) In how many ways can they line up if they must alternate woman, man, woman, man, and so on?

AG
Ankit Gupta
Numerade Educator
01:28

Problem 24

A set of reference books consists of eight volumes numbered 1 through 8 .
(a) In how many ways can the eight books be arranged so that Volume 8 is in the correct position on the shelf (i.e., the last one from left to right)?
(b) In how many ways can the eight books be arranged so that Volumes 1 and 2 are in their correct positions on the shelf?

Kyler Gray
Kyler Gray
Numerade Educator
02:46

Problem 25

Determine the number of outcomes $N$ in each sample space.
(a) A coin is tossed 10 times in a row. The result of each toss ( $H$ or $T)$ is observed.
(b) A die is rolled four times in a row. The number that comes up on each roll is observed.
(c) A die is rolled four times in a row. The sum of the numbers rolled is observed.

Nick Johnson
Nick Johnson
Numerade Educator
01:51

Problem 26

Determine the number of outcomes $N$ in each sample spac
(a) A student randomly answers a 10-question true-fals quiz. The student's answer $(T$ or $F)$ to each question observed.
(b) A student randomly answers a 10-question multipl choice quiz. The student's answer $(A, B, C, D,$ or $E)$ each question is observed.

Joshua Eastwood
Joshua Eastwood
Numerade Educator
00:50

Problem 27

The board of directors of the XYZ Corporation has 15 members.
(a) How many different slates of four officers (a President, a Vice President, a Treasurer, and a Secretary) can be chosen?
(b) A four-person committee needs to be selected to conduct a search for a new CEO. In how many ways can the search committee be selected?

Heather Zimmers
Heather Zimmers
Numerade Educator
01:06

Problem 28

There are 10 athletes entered in an Olympic event
(a) In how many ways can one pick the winne gold, silver, and bronze medals?
(b) In how many ways can one pick the seven ath will not earn any medals?

AG
Ankit Gupta
Numerade Educator
00:21

Problem 29

The Brute Force Bandits is a punk rock band planning their next concert tour. The band has a total of 30 new songs in their repertoire.
(a) How many different set lists of 18 songs can the band select to play on the tour? (Hint: Assume that the order in which the songs are listed is irrelevant.)
(b) How many different ways are there for the band to record a CD consisting of 18 songs chosen from the 30 new songs? (Hint: In recording a CD, the order in which the songs appear on the $\mathrm{CD}$ is relevant.)

Tony Ni
Tony Ni
Numerade Educator
01:46

Problem 30

There are 347 NCAA Division I college basketball teams.
(a) How many different top-25 rankings are possible? [Assume that every team has a chance to be a top- 25 team.]
(b) How many ways are there to choose 64 teams (unseeded) to make it to the NCAA tournament? [Assume every combination of 64 teams is possible.

James Kiss
James Kiss
Numerade Educator
01:46

Problem 30

There are 347 NCAA Division I college basketball teams.
(a) How many different top-25 rankings are possible? [Assume that every team has a chance to be a top-25 team.
(b) How many ways are there to choose 64 teams (unseeded) to make it to the NCAA tournament? [Assume every combination of 64 teams is possible.]

James Kiss
James Kiss
Numerade Educator
01:53

Problem 31

A major league baseball team roster consists of 40 players of which 25 are considered active.
(a) How many ways are there for a manager to select 25 active players from a major league roster?
(b) How many ways can a manager select a nine-player batting lineup from the active roster for opening day?

Clarissa Noh
Clarissa Noh
Numerade Educator
00:33

Problem 32

Bob has 20 different dress shirts in his wardrobe.
(a) In how many ways can Bob select seven shirts to pack for a business trip?
(b) In how many ways can Bob select 5 of the 7 dress shirts he packed for the business trip $-$ one for the Monday meeting, one for the Tuesday dinner, one for the Wednesday party, one for the Thursday conference, and one for the Friday date?

Ashley Volpe
Ashley Volpe
Numerade Educator
02:04

Problem 33

Consider the sample space $S=\left\{o_{1}, o_{2}, o_{3}, o_{4}, o_{5}\right\} .$ Suppose that $\operatorname{Pr}\left(o_{1}\right)=0.22$ and $\operatorname{Pr}\left(o_{2}\right)=0.24$
(a) Find the probability assignment for the probability space when $o_{3}, o_{4},$ and $o_{5}$ all have the same probability.
(b) Find the probability assignment for the probability space when $\operatorname{Pr}\left(o_{5}\right)=0.1$ and $o_{3}$ has the same probability as $o_{4}$ and $o_{5}$ combined.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:28

Problem 34

Consider the sample space $S=\left\{o_{1}, o_{2}, o_{3}, o_{4}\right\} .$ Suppose that $\operatorname{Pr}\left(o_{1}\right)+\operatorname{Pr}\left(o_{2}\right)=\operatorname{Pr}\left(o_{3}\right)+\operatorname{Pr}\left(o_{4}\right)$ and that $\operatorname{Pr}\left(o_{1}\right)=0.15$
(a) Find the probability assignment for the probability space when $o_{2}$ and $o_{3}$ have the same probability.
(b) Find the probability assignment for the probability space when $\operatorname{Pr}\left(o_{3}\right)=0.22$

Lucas Finney
Lucas Finney
Numerade Educator
02:48

Problem 35

35. Four candidates are running for mayor of Happyville. According to the polls candidate $A$ has a "one in five" probability of winning [i.e., $\operatorname{Pr}(A)=1 / 5] .$ Of the other three candidates, all we know is that candidate $C$ is twice as likely to win as candidate $B$ and that candidate $D$ is three times as likely to win as candidate $B$. Find the probability assignment for this probability space.

Sonam Khatri
Sonam Khatri
Numerade Educator
01:52

Problem 36

Seven horses $(A, B, C, D, E, F,$ and $G)$ are running in the Boonsville Sweepstakes. According to the oddsmakers, $A$ has a "one in four" probability of winning [i.e., $\operatorname{Pr}(A)=1 / 4]$, $B$ has a "three in ten" probability of winning, and $C$ has a "one in twenty" probability of winning. The remaining four horses all have the same probability of winning. Find the probability assignment for the probability space.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
View

Problem 37

An honest coin is tossed three times in a row. Find the probability of each of the following events. (Hint: Do Exercise 11 first.)
(a) $E_{1}$; "the coin comes up heads exactly twice."
(b) $E_{2}:$ "all three tosses come up the same."
(c) $E_{3}:$ "exactly half of the tosses come up heads."
(d) $E_{4}$ : "the first two tosses come up tails"

Anas Venkitta
Anas Venkitta
Numerade Educator
04:42

Problem 38

A student randomly guesses the answers to a four-question true-or-false $(T-F)$ quiz. Find the probability of each of the following events. (Hint: Do Exercise 12 first.)
(a) $E_{1}:$ "the student answers $F$ on two of the four questions."
(b) $E_{2}$ : "the student answers $F$ on at least two of the four questions."
(c) $E_{3}:$ "the student answers $F$ on at most two of the four questions."
(d) $E_{4}$ : "the student answers $F$ to the first two questions."

Derrick Hanson
Derrick Hanson
Numerade Educator
02:34

Problem 39

A pair of honest dice is rolled. Find the probability of each of the following events. (Hint: Do Exercise 13 first.)
(a) $E_{1}$ : "pairs are rolled." (A "pair" is a roll in which both dice come up the same number.)
(b) $E_{2}:$ "craps are rolled." ("Craps" is a roll in which the sum of the two numbers rolled is $2,3,$ or $12 .)$
(c) $E_{3}:$ "a natural is rolled." (A "natural" is a roll in which the sum of the two numbers rolled is 7 or $11 .$ )

Kerry Thornton-Genova
Kerry Thornton-Genova
Numerade Educator
02:27

Problem 40

A card is drawn at random out of a well-shuffled deck of 52 cards. Find the probability of each of the following events. (Hint: Do Exercise 14 first.)
(a) $E_{1}:$ "draw a queen."
(b) $E_{2}:$ "draw a heart."
(c) $E_{3}:{ }^{*}$ draw a face card." $\left(\mathrm{A}^{\text {" }}\right.$ face" card is a jack, queen, or king.)

AG
Ankit Gupta
Numerade Educator
01:53

Problem 42

Five candidates $(A, B, C, D,$ and $E)$ have a chance to be selected to be on American Idol. Any subset of them (including none of them or all of them ) can be selected, and assume that the selection process is completely random (the subsets of candidates are all equally likely). Find the probability of each of the following events. (Hint: Do Exercise 16 first.)
(a) $E_{1}:{ }^{*}$ two candidates get selected."
(b) $E_{2}:$ "three candidates get selected."
(c) $E_{3}:{ }^{4}$ three candidates get selected, and $A$ is not one of them."

Aman Gupta
Aman Gupta
Numerade Educator
01:51

Problem 43

A student takes a 10 -question true-or-false quiz and randomly guesses the answer to each question. Suppose that a correct answer is worth 1 point and an incorrect answer is worth -0.5 points. Find the probability that the student
(a) gets 10 points.
(b) gets -5 points.
(c) gets 8.5 points.
(d) gets 8 or more points.
(e) gets 5 points.
(f) gets 7 or more points.

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:32

Problem 44

Suppose that the probability of giving birth to a boy and the probability of giving birth to a girl are both $0.5 .$ Find the probability that in a family of four children,
(a) all four children are girls.
(b) there are two girls and two boys.
(c) the youngest child is a girl.
(d) the oldest child is a boy.

AG
Ankit Gupta
Numerade Educator
01:49

Problem 45

The Tasmania State University glee club has 15 members. A quartet of four members must be chosen to sing at the university president's reception. Assume that the quartet is chosen randomly by drawing the names out of a hat. Find the probability that
(a) Alice (one of the members of the glee club) is chosen to be in the quartet.
(b) Alice is not chosen to be in the quartet.
(c) the four members chosen for the quartet are Alice, Bert, Cathy, and Dale.

Jake Zanazzi
Jake Zanazzi
Numerade Educator
03:04

Problem 46

Ten professional basketball teams are participating in a draft lottery. (A draft lottery is a lottery to determine the order in which teams get to draft players.) Ten balls, each containing the name of one team (call them $A, B, C, D, E$, $F, G, H, I,$ and $J$ for short), are placed in an urn and thoroughly mixed. Four balls are drawn, one at a time, from the urn. The four teams chosen get to draft first and in the order they are chosen. The remaining six teams have to draft in reverse order of season records. Find the probability that
(a) $A$ is the first team chosen.
(b) $A$ is one of the four teams chosen.
(c) $A$ is not one of the four teams chosen.

Jeff Vermeire
Jeff Vermeire
Numerade Educator
04:13

Problem 47

An honest coin is tossed 10 times in a row. The result of each toss ( $H$ or $T$ ) is observed. Find the probability of the event $E="$ a $T$ comes up at least once." (Hint: Find the probability of the complementary event.)

AG
Ankit Gupta
Numerade Educator
01:36

Problem 48

Imagine a game in which you roll an honest die three times. Find the probability of the event $E="$ at least one of the rolls of the dice comes up a 6." (Hint: See Example 16.22.)

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
00:40

Problem 49

Find the odds of each of the following events.
(a) an event $E$ with $\operatorname{Pr}(E)=4 / 7$
(b) an event $E$ with $\operatorname{Pr}(E)=0.6$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
04:38

Problem 50

Find the odds of each of the following events.
(a) an event $E$ with $\operatorname{Pr}(E)=3 / 11$
(b) an event $E$ with $\operatorname{Pr}(E)=0.375$

Chris Wojturski
Chris Wojturski
Numerade Educator
00:59

Problem 51

In each case, find the probability of an event $E$ having the given odds.
(a) The odds in favor of $E$ are 3 to 5 .
(b) The odds against $E$ are 8 to 15 .

Ashly Sunny
Ashly Sunny
Numerade Educator
02:42

Problem 52

In each case, find the probability of an event $E$ having the given odds.
(a) The odds in favor of $E$ are 4 to 3 .
(b) The odds against $E$ are 12 to 5 .
(c) The odds in favor of $E$ are the same as the odds against $E$

Georgiann Andersen
Georgiann Andersen
Numerade Educator
01:43

Problem 53

The scoring for a Psych 101 final grade is shown in Table 16-18. The last row of the table shows Paul's individual scores. Find Paul's score in the course, expressed as a percent.
$$\begin{array}{l|c|c|c|c|c|c} & \text { Test 1 } & \text { Test 2 } & \text { Test 3 } & \text { Quizzes } & \text { Paper } & \text { Final } \\\hline \text { Weight } & 15 \% & 15 \% & 15 \% & 10 \% & 25 \% & 20 \% \\\hline \begin{array}{l}\text { Points } \\\text { possible }\end{array} & 100 & 100 & 100 & 120 & 100 & 180 \\\hline \begin{array}{l}\text { Paul's } \\\text { score }\end{array} & 77 & 83 & 91 & 90 & 87 & 144\end{array}$$

Linh Vu
Linh Vu
Numerade Educator
00:38

Problem 54

Table $16-19$ shows the aggregate scores of a golf player over an entire tournament.\begin{array}{l|r|r|r|r|r}\text { Score } & 2 & 3 & 4 & 5 & 6 \\\hline\begin{array}{l}\text { Percentage } \\\text { of holes }\end{array} & 1.4 \% & 36.1 \% & 50 \% & 11.1 \% & 1.4 \%\end{array}

Ian Maurer
Ian Maurer
Numerade Educator
01:55

Problem 55

At Thomas Jefferson High School, the student body is divided by age as follows: $7 \%$ of the students are $14,22 \%$ of the students are $15,24 \%$ of the students are $16,23 \%$ of the students are $17,19 \%$ of the students are 18 , and the rest of the students are $19 .$ Find the average age of the students at Thomas Jefferson High School.

Victor Salazar
Victor Salazar
Numerade Educator
03:41

Problem 56

In 2005 the Middletown $Z$ oo averaged 4000 visitors on sunny days, 3000 visitors on cloudy days, 1500 visitors on rainy days, and only 100 visitors on snowy days. The percentage of days of each type in 2005 is shown in Table $16-20 .$ Find the average daily attendance at the Middletown Zoo for 2(005

Ahmad Reda
Ahmad Reda
Numerade Educator
02:06

Problem 57

A box contains twenty $\$ 1$ bills, ten $\$ 5$ bills, five $\$ 10$ bills, four $\$ 20$ bills, and one $\$ 100$ bill. You blindly reach into the box and draw a bill at random. What is the expected value of your draw?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:06

Problem 58

A box contains twenty $\$ 1$ bills, ten $\$ 5$ bills, five $\$ 10$ bills, four $\$ 20$ bills, and one $\$ 100$ bill. You blindly reach into the box and draw a bill at random. What is the expected value of your draw?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
05:22

Problem 58

A basketball player shoots two consecutive free throws. Each free-throw is worth 1 point and has probability of success $p=3 / 4$. Let $X$ denote the number of points scored. Find the expected value of $X$.

Ramon Kryzhan
Ramon Kryzhan
Numerade Educator
00:34

Problem 59

A fair coin is tossed three times. Find the expected number of heads that come up.

Trinity Steen
Trinity Steen
Numerade Educator
02:59

Problem 60

A pair of honest dice is rolled once. Find the expected value of the sum of the two numbers rolled. (Hint: See Example 16.20 ).

Gus Steppen
Gus Steppen
Numerade Educator
01:53

Problem 61

Suppose that you roll a pair of honest dice. If you roll a total of $7,$ you win $\$ 18$; if you roll a total of 11 , you win $\$ 54$; if you roll any other total, you lose $\$ 9 .$ Find the expected payoff for this game.

AG
Ankit Gupta
Numerade Educator
00:36

Problem 62

On an American roulette wheel, there are 38 numbers:
$00,0,1,2, \ldots, 36 .$ If you bet $\$ N$ on any one number-say. for example, on 10 - you win $\$ 36 N$ if 10 comes up (i.e., you get $\$ 37 N$ back - your original bet plus your $\$ 36 N$ profit $) ;$ if any other number comes up, you lose your $\$ N$ bet. Find the expected payoff of a $\$ 1$ bet on 10 (or any other number).

James Macpherson
James Macpherson
Numerade Educator
03:26

Problem 64

Suppose that you roll a single die. If an odd number (1,3,0
5) comes up, you win the amount of your roll (\$1, \$3, or \$5 respectively). If an even number $(2,4,$ or 6$)$ comes up, you have to pay the house the amount of your roll $(\$ 2, \$ 4,$ or $\$ 6$ respectively).
(a) Find the expected payoff for this game.
(b) Is this a fair game? Explain.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
04:51

Problem 65

Joe is buying a new plasma TV at Circuit Town. The salesman offers Joe a three-year extended warranty for $\$ 80 .$ The salesman tells Joe that $24 \%$ of these plasma TVs require repairs within the first three years, and the average cost of a repair is \$400. Should Joe buy the extended warranty? Explain your reasoning.

Jon Southam
Jon Southam
Numerade Educator
00:52

Problem 66

Jackie is buying a new MP3 player from Better Buy. The store offers her a two-year extended warranty for $\$ 19 .$ Jackie read in a consumer magazine that for this model MP3, $5 \%$ require repairs within the first two years at an average cost of \$50. Should Jackie buy the extended warranty? Explain your reasoning.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
04:02

Problem 67

The service history of the Prego SUV is as follows: $50 \%$ will need no repairs during the first year, $35 \%$ will have repair costs of $\$ 500$ during the first year, $12 \%$ will have repair costs of $\$ 1500$ during the first year, and the remaining SUVs (the real lemons) will have repair costs of $\$ 4000$ during their first year. Determine the price that the insurance company should charge for a one-year extended warranty on a Prego SUV if it wants to make an average profit of $\$ 50$ per policy.

Aman Gupta
Aman Gupta
Numerade Educator
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Problem 68

An insurance company plans to sell a $\$ 250,000$ one-year term life insurance policy to a 60 -year-old male. Of 2.5 million men having similar risk factors, the company estimates that 7500 of them will die in the next year. What is the premium that the insurance company should charge if it would like to make a profit of $\$ 50$ on each policy?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:43

Problem 69

The ski club at Tasmania State University has 35 members (15 females and 20 males). A committee of three members - a President, a Vice President, and a Treasurer must be chosen.
(a) How many different three-member committees can be chosen?
(b) How many different three-member committees can be chosen in which the committee members are all females?
(c) How many different three-member committees can be chosen in which the committee members are all the same gender?
(d) How many different three-member committees can be chosen in which the committee members are not all the same gender?

Heather Zimmers
Heather Zimmers
Numerade Educator
02:43

Problem 70

The ski club at Tasmania State University has 35 members (15 females and 20 males). A committee of three members - a President, a Vice President, and a Treasurer must be chosen.
(a) How many different three-member committees can be chosen?
(b) How many different three-member committees can be chosen in which the committee members are all females?
(c) How many different three-member committees can be chosen in which the committee members are all the same gender?
(d) How many different three-member committees can be chosen in which the committee members are not all the same gender?

Heather Zimmers
Heather Zimmers
Numerade Educator
08:16

Problem 71

Andy and Roger are playing in a tennis match. (A tennis match is a best-of-five contest: The first player to win three games wins the match, and there are no ties.) We can describe the outcome of the tennis match by a string of letters $(A$ or $R$ ) that indicate the winner of each game. For example, the string $R A R R$ represents an outcome where Roger wins games $1,3,$ and $4,$ at which point the match is over (game 5 is not played).
(a) Describe the event "Roger wins the match in game $5 . "$
(b) Describe the event "Roger wins the match."
(c) Describe the event "the match goes five games."

Christine Anacker
Christine Anacker
Numerade Educator
02:47

Problem 72

Two teams (call them $X$ and $Y$ ) play against each other in the World Series. (The World Series is a best-of-seven series: The first team to win four games wins the series, and games cannot end in a tic.) We can describe an outcome for the World Series by writing a string of letters that indicate (in order) the winner of each game. For example, the string $X Y X X Y X$ represents the outcome: $X$ wins game $1, Y$ wins game $2, X$ wins game $3,$ and so on.
(a) Describe the event $" X$ wins in five games"
(b) Describe the event "the series is over in game $5 . "$
(c) Describe the event "the series is over in game 6 "
(d) Find the size of the sample space.

Joe Lesueur
Joe Lesueur
Numerade Educator
02:46

Problem 73

An urn contains seven red balls and three blue balls.
(a) If three balls are selected all at once, what is the probability that two are blue and one is red?
(b) If three balls are selected by pulling out a ball, noting its color, and putting it back in the urn before the next selection, what is the probability that only the first and third balls drawn are blue?
(c) If three balls are selected one at a time without putting them back in the urn, what is the probability that only the first and third balls drawn are blue?

Julian Wong
Julian Wong
Numerade Educator
00:45

Problem 74

If we toss an honest coin 10 times, what is the probability of
(a) getting 5 heads and 5 tails?
(b) getting 3 heads and 7 tails?

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
03:19

Problem 75

If an honest coin is tossed $N$ times, what is the probability of getting the same number of heads as tails? (Hints: 1 . Try Exercise 74 (a) first. 2. Consider two cases: $N$ even and $N$ odd.

Raj Bala
Raj Bala
Numerade Educator
03:19

Problem 75

If an honest coin is tossed $N$ times, what is the probability of getting the same number of heads as tails? (Hints: $1 .$ Try Exercise 74 (a) first. 2. Consider two cases: $N$ even and $N$ odd.)

Raj Bala
Raj Bala
Numerade Educator
01:31

Problem 76

A draw poker hand consists of 5 cards taken from a deck of 52 cards where the order of the cards is irrelevant (see Examples 16.15 and 16.23 ). Assuming you are playing with an honest deck, find the probability of each of the following hands.

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
00:41

Problem 77

There are 500 tickets sold in a raffle. If you have three of these tickets and five prizes are to be given, what is the probability that you will win at least one prize? (Give your answer in symbolic notation.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:23

Problem 78

This exercise refers to the game of chuck-a-luck discussed in Example 16.34 . Explain why, when you roll three dice,
(a) the probability of rolling two 4 's plus another number $($ not a 4$)$ is $15 / 216$
(b) the probability of rolling one 4 plus two other numbers $($ not 4 's $)$ is $75 / 216$.
(c) the probability of rolling no 4 's is $125 / 216$.

Ernest Castorena
Ernest Castorena
Numerade Educator
01:52

Problem 79

Yahtzee. Yahtzee is a dice game in which five standard dice are rolled at one time.
(a) What is the probability of scoring "Yahtzee" with one roll of the dice? (You score Yahtzee when all five dice match.)
(b) What is the probability of a four of a kind with one roll of the dice? (A four of a kind is rolled when four of the five dice match.)
(c) What is the probability of rolling a large straight in one roll of the dice? (A large straight consists of five numbers in succession on the dice.)

Christopher Stanley
Christopher Stanley
Numerade Educator
03:06

Problem 80

In head-to-head, 7 -card stud poker you make your hand by selecting your 5 best cards from the 2 in your hand and 3 from the 5 common cards showing on the table (the "flush draw") $-$ if the last card ("river" card) is a spade you will have an ace high flush and a guaranteed win. Assume that your opponent has a decent hand and if you don't get the spade on the river card you will lose the hand.
(a) Suppose there is $\$ 100$ in the pot and your opponent moves "all-in" with a $\$ 50$ bet. Should you call the bet or fold? Explain.
(b) Suppose there is $\$ 100$ in the pot and your opponent moves "all-in" with a $\$ 20$ bet. Should you call the bet or fold? Explain.

Victoria Dollar
Victoria Dollar
Numerade Educator