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  • Elementary Statistics a Step by Step Approach
  • Probability and Counting Rules

Elementary Statistics a Step by Step Approach

Allan G. Bluman

Chapter 4

Probability and Counting Rules - all with Video Answers

Educators

+ 27 more educators

Section 1

Sample Spaces and Probability

02:09

Problem 1

What is a probability experiment?

Richard Miller
Richard Miller
Numerade Educator
02:10

Problem 2

Define sample space.

Souvik Ghosh
Souvik Ghosh
Numerade Educator
01:44

Problem 3

What is the difference between an outcome and an event?

Richard Miller
Richard Miller
Numerade Educator
02:09

Problem 4

What are equally likely events?

Richard Miller
Richard Miller
Numerade Educator
00:54

Problem 5

What is the range of the values of the probability of an event?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:00

Problem 6

When an event is certain to occur, what is its probability?

Richard Miller
Richard Miller
Numerade Educator
00:46

Problem 7

If an event cannot happen, what value is assigned to its probability?

Richard Miller
Richard Miller
Numerade Educator
01:14

Problem 8

What is the sum of the probabilities of all the outcomes in a sample space?

Richard Miller
Richard Miller
Numerade Educator
00:48

Problem 9

If the probability that it will rain tomorrow is $0.20,$ what is the probability that it won't rain tomorrow? Would you recommend taking an umbrella?

Richard Miller
Richard Miller
Numerade Educator
01:19

Problem 10

A probability experiment is conducted. Which of these cannot be considered a probability outcome?
$$
\begin{array}{llll}{\text { a. } \frac{2}{3}} & {\text { d. } 1.65} & {g .1} \\ {\text { b. } 0.63} & {e .-0.44} & {h .125 \%} \\ {\text { c. }-\frac{3}{5}} & {\text { f. } 0} & {\text { i. } 24 \%}\end{array}
$$

Richard Miller
Richard Miller
Numerade Educator
02:23

Problem 11

Classify each statement as an example of classical probability, empirical probability, or subjective probability.
a. The probability that a person will watch the 6 o'clock evening news is 0.15 .
b. The probability of winning at a Chuck-a-Luck game is $\frac{5}{36}$.
c. The probability that a bus will be in an accident on a specific run is about $6 \% .$
d. The probability of getting a royal flush when five cards are selected at random is $\frac{1}{649,740} .$

Richard Miller
Richard Miller
Numerade Educator
01:40

Problem 12

Classify each statement as an example of classical probability, empirical probability, or subjective probability.
a. The probability that a student will get a C or better in a statistics course is about $70 \% .$
b. The probability that a new fast-food restaurant will be a success in Chicago is $35 \% .$
c. The probability that interest rates will rise in the next 6 months is 0.50 .
d. The probability that the unemployment rate will fall next month is 0.03 .

Richard Miller
Richard Miller
Numerade Educator
01:20

Problem 13

Rolling a Die If a die is rolled one time, find these probabilities:
a. Getting a 7
b. Getting an odd number
c. Getting a number less than 7
d. Getting a prime number ( 2 , 3, or 5)

Richard Miller
Richard Miller
Numerade Educator
01:37

Problem 14

Rolling a Die If a die is rolled one time, find these probabilities:
a. Getting a number less than 7.
b. Getting a number greater than or equal to 3
c. Getting a number greater than 2 and an even number
d. Getting a number less than 1

Richard Miller
Richard Miller
Numerade Educator
02:50

Problem 15

Rolling Two Dice If two dice are rolled one time, find the probability of getting these results:
a. A sum of 5
b. A sum of 9 or 10
c. Doubles

Richard Miller
Richard Miller
Numerade Educator
01:27

Problem 16

Rolling Two Dice If two dice are rolled one time, find the probability of getting these results:
a. A sum less than 9
b. A sum greater than or equal to 10
c. A 3 on one die or on both dice.

Richard Miller
Richard Miller
Numerade Educator
02:08

Problem 17

Drawing a Card If one card is drawn from a deck, find the probability of getting these results:
a. An ace
b. A heart
c. A 6 of spades
d. A 10 or a jack
e. A card whose face values less than 7 (Count aces as 1.)

Richard Miller
Richard Miller
Numerade Educator
01:38

Problem 18

Drawing a Card If a card is drawn from a deck, find the probability of getting these results:
a. A 6 and a spade
b. A black king
c. A red card and a 7
d. A diamond or a heart
e. A black card

Richard Miller
Richard Miller
Numerade Educator
02:54

Problem 19

Shopping Mall Promotion A shopping mall has set up a promotion as follows. With any mall purchase of $$\$ 50$$ or more, the customer gets to spin the wheel shown here. If a number 1 comes up, the customer wins $$\$ 10$$. If the number 2 comes up, the customer wins $$\$ 5 ;$$ and if the number 2 comes up, the customer wins a discount coupon. Find the following probabilities.
a. The customer wins $$\$ 10$$.
b. The customer wins money.
c. The customer wins a coupon.
(Figure can't copy)

Abdul Vahid M
Abdul Vahid M
Numerade Educator
01:09

Problem 20

Selecting a State Choose one of the 50 states at random.
a. What is the probability that it begins with the letter M?
b. What is the probability that it doesn't begin with a vowel?

Richard Miller
Richard Miller
Numerade Educator
06:50

Problem 21

Human Blood Types Human blood is grouped into four types. The percentages of Americans with each type are listed below.
$$
\begin{array}{ccccccc}{0} & {43 \%} & {\text { A } 40 \%} & {\text { B } 12 \%} & {} & {\text { AB } 5 \%}\end{array}
$$
Choose one American at random. Find the probability that this person
a. Has type B blood
b. Has type AB or O blood
c. Does not have type O blood

Mico Abunda
Mico Abunda
Numerade Educator
01:27

Problem 22

2014 Top Albums (Based on U.S. sales) Of all of the U.S. album sales 1989 (Taylor Swift) accounted for $25 \%$ of sales, Frozen (Various Artists) accounted for $24.1 \%$ of sales, In the Lonely Hour (Sam Smith) accounted for $8.2 \%$ of sales. What is the probability that a randomly selected album was something other than these three albums?

Richard Miller
Richard Miller
Numerade Educator
02:24

Problem 23

Prime Numbers A prime number is a number that is evenly divisible only by 1 and itself. The prime numbers less than 100 are listed below.
$$
\begin{array}{lllllllllll}{2} & {3} & {5} & {7} & {11} & {13} & {17} & {19} & {23} & {29} & {31} \\ {37} & {41} & {43} & {47} & {53} & {59} & {61} & {67} & {71} & {73} & {79} \\ {83} & {89} & {97} & {} & {} & {} & {} & {}\end{array}
$$
Choose one of these numbers at random. Find the probability that
a. The number is odd
b. The sum of the digits is odd
c. The number is greater than 70

EA
Emmanuel Arhin
Numerade Educator
02:54

Problem 24

Rural Speed Limits Rural speed limits for all 50 states are indicated below.
(Table can't copy)
Choose one state at random. Find the probability that its speed limit is
a. 60 or 70 miles per hour
b. Greater than 65 miles per hour
c. 70 miles per hour or less

Abdul Vahid M
Abdul Vahid M
Numerade Educator
08:35

Problem 25

Gender of Children A couple has 4 children. Find each probability.
a. All girls
b. Exactly two girls and two boys
c. At least one child who is a girl
d. At least one child of each gender

TH
Timothy Hollman
Numerade Educator
01:10

Problem 26

Sources of Energy Uses in the United States A break- down of the sources of energy used in the United States is shown below. Choose one energy source at random. Find the probability that it is
a. Not oil
b. Natural gas or oil
c. Nuclear
$$
\begin{array}{lll}{\text { Oil } 39 \%} & {\text { Natural gas } 24 \%} & {\text { Coal } 23 \%} \\ {\text { Nuclear } 8 \%} & {\text { Hydropower } 3 \%} & {\text { Other } 3 \%}\end{array}
$$

Richard Miller
Richard Miller
Numerade Educator
01:13

Problem 27

Game of Craps In a game of craps, a player wins on the first roll if the player rolls a sum of 7 or $11,$ and the player loses if the player rolls a $2,3,$ or $12 .$ Find the probability that the game will last only one roll.

Richard Miller
Richard Miller
Numerade Educator
06:27

Problem 28

Computers in Elementary Schools Elementary and secondary schools were classified by the number of computers they had.
$$
\begin{array}{l|llllll}{\text { Computers }} & {1-10} & {11-20} & {21-50} & {51-100} & {100+} \\ \hline \text { Schools } & {3170} & {4590} & {16,741} & {23,753} & {34,803}\end{array}
$$
Choose one school at random. Find the probability that it has
a. 50 or fewer computers
b. More than 100 computers
c. No more than 20 computers

Hannah Wilds
Hannah Wilds
Numerade Educator
01:54

Problem 29

College Debt The following information shows the amount of debt students who graduated from college incur for a specific year.
$$
\begin{array}{cccc}{\$ 1 \text { to }} & {\$ 5001 \text { to }} & {\$ 20,001 \text { to }} & {} \\ {\$ 5000} & {\$ 20,000} & {\$ 50,000} & {\$ 50,000+} \\ {27 \%} & {40 \%} & {19 \%} & {14 \%}\end{array}
$$
If a person who graduates has some debt, find the probability that
$$
\begin{array}{l}{\text { a. It is less than } \$ 5001} \\ {\text { b. It is more than } \$ 20,000} \\ {\text { c. It is between } \$ 1 \text { and } \$ 20,000} \\ {\text { d. It is more than } \$ 50,000}\end{array}
$$

Richard Miller
Richard Miller
Numerade Educator
01:00

Problem 30

Population of Hawaii The population of Hawaii is $22.7 \%$ white, $1.5 \%$ African-American, $37.7 \%$ Asian, $0.2 \%$ Native American/Alaskan, $9.46 \%$ Native Hawaiian/Pacific Islander, $8.9 \%$ Hispanic, $19.4 \%$ two or more races, and $0.14 \%$ some other. Choose one Hawaiian resident at random. What is the probability that he/she is a Native Hawaiian or Pacific Islander? Asian? White?

Richard Miller
Richard Miller
Numerade Educator
01:39

Problem 31

Crimes Committed The numbers show the number of crimes committed in a large city. If a crime is selected at random, find the probability that it is a motor vehicle theft. What is the probability that it is not an assault?
$$
\begin{array}{ll}{\text { Theft }} & {1375} \\ {\text { Burglary of home or office }} & {500} \\ {\text { Motor vehicle theft }} & {275} \\ {\text { Assault }} & {200} \\ {\text { Robbery }} & {125} \\ {\text { Rape or homicide }} & {25}\end{array}
$$

Richard Miller
Richard Miller
Numerade Educator
02:24

Problem 32

Living Arrangements for Children Here are the living arrangements of children under 18 years old living in the United States in a recent year. Numbers are in thousands.
$$
\begin{array}{lc}{\text { Both parents }} & {51,823} \\ {\text { Mother only }} & {17,283} \\ {\text { Father only }} & {2,572} \\ {\text { Neither parent }} & {3,041}\end{array}
$$
Choose one child at random; what is the probability that the child lives with both parents? With the mother present?

Richard Miller
Richard Miller
Numerade Educator
01:55

Problem 33

Motor Vehicle Accidents During a recent year, there were 13.5 million automobile accidents, 5.2 million truck accidents, and $178,000$ motorcycle accidents. If one accident is selected at random, find the probability
that it is either a truck or motorcycle accident. What is the probability that it is not a truck accident?

Richard Miller
Richard Miller
Numerade Educator
00:39

Problem 34

Federal Government Revenue The source of federal
government revenue for a specific year is
$50 \%$ from individual income taxes
$32 \%$ from social insurance payroll taxes
$10 \%$ from corporate income taxes
$3 \%$ from excise taxes
$5 \%$ other
If a revenue source is selected at random, what is the probability that it comes from individual or corporate income taxes?

Richard Miller
Richard Miller
Numerade Educator
02:10

Problem 35

$$
\begin{array}{l}{\text { Selecting a Bill A box contains a } \$ 1 \text { bill, a } \$ 5 \text { bill, a }} \\ {\$ 10 \text { bill, and a } \$ 20 \text { bill. A bill is selected at random, and }} \\ {\text { it is not replaced; then a second bill is selected at random. }} \\ {\text { Draw a tree diagram and determine the sample space. }}\end{array}
$$

Richard Miller
Richard Miller
Numerade Educator
01:57

Problem 36

Tossing Coins Draw a tree diagram and determine the sample space for tossing four coins.

Richard Miller
Richard Miller
Numerade Educator
01:32

Problem 37

Selecting Numbered Balls Four balls numbered 1 through 4 are placed in a box. A ball is selected at random, and its number is noted; then it is replaced. A second ball is selected at random, and its number is noted. Draw a tree diagram and determine the sample space.

Richard Miller
Richard Miller
Numerade Educator
01:07

Problem 38

Family Dinner Combinations A family special at a neighborhood restaurant offers dinner for four for $\$ 39.99$. There are 3 appetizers available, 4 entrees, and 3 desserts from which to choose. The special includes one of each. Represent the possible dinner combinations with a tree diagram.

Richard Miller
Richard Miller
Numerade Educator
01:12

Problem 39

Required First-Year College Courses First-year students at a particular college must take one English class, one class in mathematics, a first-year seminar, and an elective. There are 2 English classes to choose from, 3 mathematics classes, 5 electives, and everyone takes the same first-year seminar. Represent the possible schedules, using a tree diagram.

Richard Miller
Richard Miller
Numerade Educator
01:54

Problem 40

Tossing a Coin and Rolling a Die A coin is tossed; if it falls heads up, it is tossed again. If it falls tails up, a die is rolled. Draw a tree diagram and determine the outcomes.

Daniel Faulkenberry
Daniel Faulkenberry
Numerade Educator
02:19

Problem 41

Distribution of CEO Ages The distribution of ages of CEOs is as follows:
$$
\begin{array}{ll}{\text { Age }} & {\text { Frequency }} \\ \hline 21-30 &\quad\quad { 1 } \\ {31-40} &\quad\quad {8} \\ {41-50} &\quad \quad{27} \\ {51-60} & \quad\quad{29} \\ {61-70} &\quad\quad {24} \\ {71-4 p} & \quad\quad{11}\end{array}
$$
If a CEO is selected at random, find the probability that his or her age is
a. Between 31 and 40
b. Under 31
c. Over 30 and under 51
d. Under 31 or over 60

Richard Miller
Richard Miller
Numerade Educator
01:04

Problem 42

Tossing a Coin A person flipped a coin 100 times and obtained 73 heads. Can the person conclude that the coin was unbalanced?

Richard Miller
Richard Miller
Numerade Educator
00:22

Problem 43

Medical Treatment A medical doctor stated that with a certain treatment, a patient has a 50% chance of recovering without surgery. That is, “Either he will get well or he won’t get well.” Comment on this statement.

Richard Miller
Richard Miller
Numerade Educator
06:06

Problem 44

Wheel Spinner The wheel spinner shown here is spun twice. Find the sample space, and then determine the probability of the following events.
(Figure can't copy)
a. An odd number on the first spin and an even number on the second spin (Note: 0 is considered even.)
b. A sum greater than 4
c. Even numbers on both spins
d. A sum that is odd
e. The same number on both spins

Abdul Vahid M
Abdul Vahid M
Numerade Educator

Problem 45

Tossing Coins Toss three coins 128 times and record the number of heads (0, 1, 2, or 3); then record your results with the theoretical probabilities. Compute the
empirical probabilities of each.

Check back soon!

Problem 46

Tossing Coins Toss two coins 100 times and record the number of heads $(0,1,2) .$ Compute the probabilities of each outcome, and compare these probabilities with the theoretical results.

Check back soon!
03:47

Problem 47

$$
\begin{array}{l}{\text { Odds Odds are used in gambling games to make them }} \\ {\text { fair. For example, if you rolled a die and won every }} \\ {\text { time you rolled a } 6, \text { then you would win on average }} \\ {\text { once every } 6 \text { times. So that the game is fair, the odds of }} \\ {5 \text { to } 1 \text { are given. This means that if you bet } \$ 1 \text { and won, }} \\ {\text { you could win } \$ 5 . \text { On average, you would win } \$ 5 \text { once }} \\ {\text { in } 6 \text { rolls and lose } \$ 1 \text { on the other } 5 \text { rolls-hence the }} \\ {\text { term fair game. }}\end{array}
$$
$$
\begin{array}{l}{\text { In most gambling games, the odds given are not fair. }} \\ {\text { For example, if the odds of winning are really } 20 \text { to } 1,} \\ {\text { the house might offer } 15 \text { to } 1 \text { in order to make a profit. }} \\ {\text { Odds can be expressed as a fraction or as a ratio, }} \\ {\text { such as } \frac{5}{1}, 5: 1, \text { or } 5 \text { to } 1 . \text { Odds are computed in favor }} \\ {\text { of the event or against the event. The formulas for }} \\ {\text { odds are }}\end{array}
$$
$$
\begin{array}{l}{\text { Odds in favor }=\frac{P(E)}{1-P(E)}} \\ {\text { Odds against }=\frac{P(\bar{E})}{1-P(\bar{E})}}\end{array}
$$
In the die example,
$$
\begin{array}{c}{\text { Odds in favor of a } 6=\frac{\frac{1}{6}}{\frac{5}{6}}=\frac{1}{5} \text { or } 1: 5} \\ {\text { Odds against a } 6=\frac{\frac{5}{6}}{\frac{1}{6}}=\frac{5}{1} \text { or } 5: 1}\end{array}
$$
Find the odds in favor of and against each event.
a. Rolling a die and getting a 2
b. Rolling a die and getting an even number
c. Drawing a card from a deck and getting a spade
d. Drawing a card and getting a red card
e. Drawing a card and getting a queen
f. Tossing two coins and getting two tails
g. Tossing two coins and getting exactly one tail

Richard Miller
Richard Miller
Numerade Educator

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