Question 1

Find the indicated probability.
-If you flip a coin three times, the possible outcomes are 
HHH HHT HTH HTT THH THT TTH TTT. What is the probability of getting at least one head?  
A)$\frac { 7 } { 8 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 1 } { 2 }$

Accepted Answer

The probability of getting at least one head is found by subtracting the probability of getting no heads (TTT) from 1. Since there is only one way to get no heads out of 8 possible outcomes, the probability of getting at least one head is $1 - \frac{1}{8} = \frac{7}{8}$.

Question 2

Determine whether the events are independent.
-When a coin is tossed three times, eight equally likely outcomes are possible. 
HHH HHT HTH HTT
THH THT TTH TTT
Let
$A =$ event the first two tosses are the same
$B =$ event the total number of heads is one.
Are $A$ and $B$ independent events?

Accepted Answer

Events $A$ and $B$ are not independent because the occurrence of event $A$ affects the probability of event $B$ occurring. If the first two tosses are the same (event $A$), it's impossible to have exactly one head in total (event $B$), since you already have either two heads or two tails from the first two tosses.

Question 3

Find the conditional probability.
-The following contingency table provides a joint frequency distribution for the popular votes cast in the 1984 presidential election by region and political party. Data are in thousands, rounded to the Nearest thousand.  A person who voted in the 1984 presidential election is selected at random. Compute the Probability that the person selected voted Democrat given that they were in the Northeast. 
A)0.098
B)0.406
C)0.241
D)0.442

Accepted Answer

The answer of Find the conditional probability.
-The following contingency table...

Question 4

List the outcomes comprising the specified event.
-In a competition, two people will be selected from four finalists to receive the first and second prizes. The prize winners will be selected by drawing names from a hat. The names of the four
Finalists are Jim, George, Helen, and Maggie. The possible outcomes can be represented as follows. 
JG JH JM GJ GH GM
HJ HG HM MJ MG MH
Here, for example, JG represents the outcome that Jim receives the first prize and George receives the second prize. The event $\mathrm { A }$ is defined as follows.
$\mathrm { A } =$ event that Helen gets first prize
List the outcomes that comprise the event (not A).  
A)JG, JH, JM, GJ, GH, GM, MJ
B) HJ, HG, HM
C) $\mathrm { JG } , \mathrm { JM } , \mathrm { GJ } , \mathrm { GM } , \mathrm { MJ } , \mathrm { MG }$
D) JG, JH, JM, GJ, GH, GM, MJ, MG, MH

Accepted Answer

The answer of List the outcomes comprising the specified event.
-In...

Question 5

Find the indicated probability by using the special addition rule.
-The age distribution of students at a community college is given below. $$\begin{array} { l c } 
\text { Age (years) } & \text { Number of students (f) } \
\hline \text { Under 21 } & 406 \
21 - 25 & 419 \
26 - 30 & 205 \
31 - 35 & 60 \
\text { Over } 35 & 26 \
\hline & 1116
\end{array}$$ A student from the community college is selected at random. Find the probability that the student is between 26 and 35 inclusive. Round approximations to three decimal places. 
A)265
B)0.054
C)0.237
D)0.184

Accepted Answer

The probability is found by adding the number of students aged 26-30 and 31-35, which is 205 + 60 = 265, and then dividing by the total number of students, 1116. So, the probability is $265 / 1116 = 0.237$.

Question 6

List the outcome(s)of the stated event.
-The odds against winning in a horse race are shown in the following table. $\begin{array}{l|rrrrrrr}
\text { Horse } & \# 1 & \# 2 & \# 3 & \# 4 & \# 5 & \# 6 & \# 7 \
\hline \text { Odds } & 5 & 16 & 1 & 19 & 10 & 19 & 1
\end{array}$

Based on these odds, which horses comprise: A = event the winning horse's number is above 4 ?  
A)Horses $\# 1$, #2, $\# 4$, #5, and $\# 6$
B) Horse $\# 7$
C) Horses $\# 5$, $\# 6$, and $\# 7$
D) Horses $\# 4 , \# 5$, $\# 6$, and $\# 7$

Accepted Answer

Horses #5, #6, and #7 are the only ones with numbers above 4, making them the correct choices for event A.

Question 7

Determine the number of outcomes that comprise the specified event.
-The number of hours needed by sixth grade students to complete a research project was recorded with the following results. $\begin{array}{cc}
\text { Hours } & \text { Number of students (f) } \
\hline 4 & 15 \
5 & 27 \
6 & 23 \
7 & 13 \
8 & 10 \
9 & 5 \
10+ & 5
\end{array}$

A student is selected at random. The events $A$ and $B$ are defined as follows.
$A =$ the event the student took at most 8 hours
$B =$ the event the student took at least 8 hours
Determine the number of outcomes that comprise the event (A & B).  
A)10
B) 20
C) 111
D) 98

Accepted Answer

Event A (at most 8 hours) includes students who took 4, 5, 6, 7, and 8 hours. Event B (at least 8 hours) includes students who took 8, 9, and 10+ hours. The intersection of A and B (A & B) is the number of students who took exactly 8 hours, which is 10.

Question 8

Find the conditional probability.
-If a single fair die is rolled, find the probability of a 5 given that the number rolled is odd. 
A)0.333
B)0.5
C)0.667
D)0.167

Accepted Answer

The numbers on a die that are odd are 1, 3, and 5. Given that the number rolled is odd, there are 3 possible outcomes. Only one of these outcomes is a 5, so the conditional probability is 1/3 or 0.333.

Question 9

Use the general multiplication rule to find the indicated probability.
-Among the contestants in a competition are 44 women and 30 men. If 5 winners are randomly selected, what is the probability that they are all men? 
A)0.13122
B)0.07432
C)0.14735
D)0.00885

Accepted Answer

The probability of selecting all men can be calculated using the combination formula for selecting r items out of n, which is C(n, r) = n! / [r!(n-r)!], and then applying the general multiplication rule. The total number of ways to select 5 winners out of 74 (44 women + 30 men) contestants is C(74, 5). The number of ways to select 5 men out of 30 is C(30, 5). The probability is then the ratio of these two, P(all men) = C(30, 5) / C(74, 5). Calculating this gives a probability of approximately 0.00885.

Question 10

Draw a Venn diagram and shade the described events.
-From a finite sample, events A, B, and C are mutually exclusive. Shade the collection A or B or C. 
A)

B)

C)

D)

Accepted Answer

The answer of Draw a Venn diagram and shade the...

Question 11

Find the indicated probability by using the general addition rule.
-Let $\mathrm { A }$ and $\mathrm { B }$ be events such that $\mathrm { P } ( \mathrm { A } ) = \frac { 1 } { 7 } , \mathrm { P } ( \mathrm { A }$ or $\mathrm { B } ) = \frac { 1 } { 6 }$, and $\mathrm { P } ( \mathrm { A }$ and $\mathrm { B } ) = \frac { 1 } { 12 }$. Determine $\mathrm { P } ( \mathrm { B } )$.  
A)$\frac { 5 } { 84 }$
B) $\frac { 3 } { 28 }$
C) $\frac { 1 } { 42 }$
D) $\frac { 11 } { 28 }$

Accepted Answer

Using the general addition rule: $ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) $. Plugging in the values: $\frac{1}{6} = \frac{1}{7} + P(B) - \frac{1}{12}$. Solving for $ P(B) $ gives $\frac{3}{28}$.

Question 12

Use words or symbols, as indicated, to describe the event.
-The Book Industry Study Group, Inc., performs sample surveys to obtain information on characteristics of book readers. A book reader is defined to be one who read one or more books in The six months prior to the survey; a non-book reader is defined to be one who read newspapers or Magazines but no books in the six months prior to the survey; a nonreader is defined to be one who Did not read a book, newspaper, or magazine in the six months prior to the survey. The following Data were obtained from a random sample of 1429 persons 16 years old and over.  How may people made less than $40,000? 
A)1429
B)313
C)304
D)1125

Accepted Answer

The answer of Use words or symbols, as indicated, to...

Question 13

Find the indicated probability.
-The following contingency table provides a joint frequency distribution for a group of retired people by career and age at retirement. 
Suppose one of these people is selected at random. Compute $\mathrm { P } \left( \mathrm { A } _ { 2 } \right.$ or $\left. \mathrm { C } _ { 3 } \right)$.  
A)$0.060$
B) $0.415$
C) $0.253$
D) $0.474$

Accepted Answer

The answer of Find the indicated probability.
-The following contingency table...

Question 14

Draw a Venn diagram and shade the described events.
-From a finite sample, events A, B, and C are non-mutually exclusive. Shade the collection A and B and C. 
A)

B)

C)

D)

Accepted Answer

The answer of Draw a Venn diagram and shade the...

Question 15

Use the general multiplication rule to find the indicated probability.
-The table below describes the smoking habits of a group of asthma sufferers. $\begin{array}{r|cccc} 
& &{\text { Light Heavy }} \
& \text { Nonsmoker } & \text { smoker } & \text { smoker } & \text { Total } \
\hline \text { Men } & 347 & 50 & 39 & 436 \
\text { Women } & 354 & 34 & 46 & 434 \
\text { Total } & 701 & 84 & 85 & 870
\end{array}$
 If two different people are randomly selected from the 870 subjects, find the probability that they are both heavy smokers. 
A)0.002010
B)0.0001384
C)0.009444
D)0.009546

Accepted Answer

The answer of Use the general multiplication rule to find...

Question 16

Find the indicated probability.
-If two balanced die are rolled, the possible outcomes can be represented as follows. 
$( 1,1 ) ( 2,1 ) ( 3,1 ) ( 4,1 ) ( 5,1 ) ( 6,1 )$
$( 1,2 ) ( 2,2 ) ( 3,2 ) ( 4,2 ) ( 5,2 ) ( 6,2 )$
$( 1,3 ) ( 2,3 ) ( 3,3 ) ( 4,3 ) ( 5,3 ) ( 6,3 )$
$( 1,4 ) ( 2,4 ) ( 3,4 ) ( 4,4 ) ( 5,4 ) ( 6,4 )$
$( 1,5 ) ( 2,5 ) ( 3,5 ) ( 4,5 ) ( 5,5 ) ( 6,5 )$
$( 1,6 ) ( 2,6 ) ( 3,6 ) ( 4,6 ) ( 5,6 ) ( 6,6 )$
Determine the probability that the sum of the dice is 3 or 9 .  
A)$\frac { 2 } { 9 }$
B) $\frac { 7 } { 36 }$
C) $\frac { 1 } { 6 }$
D) $\frac { 5 } { 36 }$

Accepted Answer

The answer of Find the indicated probability.
-If two balanced die...

Question 17

Use words or symbols, as indicated, to describe the event.
-The following contingency table provides a joint frequency distribution for a group of retired people by age at retirement and career.  How many of these people were not attorneys when they retired? 
A)44
B)690
C)535
D)165

Accepted Answer

The answer of Use words or symbols, as indicated, to...

Question 18

List the outcomes comprising the specified event.
-Three board members for a nonprofit organization will be selected from a group of five people. The board members will be selected by drawing names from a hat. The names of the five possible board Members are Allison, Betty, Charlie, Dave, and Emily. The possible outcomes can be represented as Follows. 
ABC ABD ABE ACD ACE
ADE BCD BCE BDE CDE
Here, for example, ABC represents the outcome that Allison, Betty, and Charlie are selected to be on the board. The events $A$ and $B$ are defined as follows.
$\mathrm { A } =$ event that Dave is selected
$B =$ event that Allison is selected
List the outcomes that comprise the event (A or B).  
A)$\mathrm { ABC } , \mathrm { ABD } , \mathrm { ABE } , \mathrm { ACD } , \mathrm { ACE } , \mathrm { ADE } , \mathrm { BCD } , \mathrm { BDE }$
B) $\mathrm { ABD } , \mathrm { ACD } , \mathrm { ADE }$
C) $\mathrm { ABC } , \mathrm { ABE } , \mathrm { ACE } , \mathrm { BCD } , \mathrm { BDE } , \mathrm { CDE }$
D) $\mathrm { ABC } , \mathrm { ABD } , \mathrm { ABE } , \mathrm { ACD } , \mathrm { ACE } , \mathrm { ADE } , \mathrm { BCD } , \mathrm { BDE } , \mathrm { CDE }$

Accepted Answer

The event (A or B) includes all outcomes where either Dave (D) is selected, Allison (A) is selected, or both. This includes all listed combinations except for those that lack both A and D (which are none in this case), thus including all possible outcomes listed.

Question 19

Use the general multiplication rule to find the indicated probability.
-You are dealt two cards successively (without replacement)from a shuffled deck of 52 playing cards. Find the probability that the first card is a king and the second card is a queen. 
A)0.002
B)0.154
C)0.127
D)0.006

Accepted Answer

The probability of drawing a king first is 4/52 (since there are 4 kings in a deck of 52 cards). After drawing a king, there are 51 cards left, and the probability of drawing a queen next is 4/51 (since there are 4 queens left in the deck). Multiplying these probabilities gives (4/52) * (4/51) = 1/13 * 1/51 = 1/663, which is approximately 0.006.

Question 20

Use Bayes's rule to find the indicated probability.
-Among students at one college are 3936 women and 3197 men. The following table provides relative-frequency distributions for subject major for males and females at the college. $$\begin{array} { l c c } 
\text { Major } & \begin{array} { l } 
\text { Relative frequency } \
\text { for women }
\end{array} & \begin{array} { c } 
\text { Relative frequency } \
\text { for men }
\end{array} \
\hline \text { Humanities } & 0.207 & 0.157 \
\text { Science } & 0.299 & 0.354 \
\text { Social Science } & 0.171 & 0.375 \
\text { Other } & 0.323 & 0.114
\end{array}$$ A student is selected at random from the college. Determine the probability that the student Selected is female given that he or she is a Humanities major. 
A)0.114
B)0.619
C)0.207
D)0.637

Accepted Answer

To find the probability that a randomly selected student is female given they are a Humanities major, we use Bayes's rule. First, calculate the total probability of being a Humanities major by multiplying the relative frequencies by the total number of women and men, then summing those products. Then, divide the probability of being a female Humanities major by this total probability. Calculation: $P(Female|Humanities) = \frac{0.207 \times 3936}{(0.207 \times 3936) + (0.157 \times 3197)} \approx 0.619$.

Quiz 4: Probability Concepts

Find the indicated probability. -If you flip a coin three times, the possible outcomes are HHH HHT HTH HTT THH THT TTH TTT. What is the probability of getting at least one head?

Determine whether the events are independent. -When a coin is tossed three times, eight equally likely outcomes are possible. HHH HHT HTH HTT THH THT TTH TTT Let $A =$ event the first two tosses are the same $B =$ event the total number of heads is one. Are $A$ and $B$ independent events?

Find the conditional probability. -If a single fair die is rolled, find the probability of a 5 given that the number rolled is odd.

Use the general multiplication rule to find the indicated probability. -Among the contestants in a competition are 44 women and 30 men. If 5 winners are randomly selected, what is the probability that they are all men?

Draw a Venn diagram and shade the described events. -From a finite sample, events A, B, and C are mutually exclusive. Shade the collection A or B or C.

Draw a Venn diagram and shade the described events. -From a finite sample, events A, B, and C are non-mutually exclusive. Shade the collection A and B and C.

Use words or symbols, as indicated, to describe the event. -The following contingency table provides a joint frequency distribution for a group of retired people by age at retirement and career. How many of these people were not attorneys when they retired?

Use the general multiplication rule to find the indicated probability. -You are dealt two cards successively (without replacement) from a shuffled deck of 52 playing cards. Find the probability that the first card is a king and the second card is a queen.