Question 1

Find the indicated probability.
-The following contingency table provides a joint frequency distribution for the popular votes cast in the presidential election by region and political party. Data are in thousands, rounded to the Nearest thousand.  A person who voted in the presidential election is selected at random. Compute the probability that The person selected was in the West and voted Republican.

Accepted Answer

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

Question 2

List the outcomes comprising the specified event.
-When a quarter is tossed four times, 16 outcomes are possible. $\begin{array} { l l l l } \text { HHHH } & \text { HHHT } & \text { HHTH } & \text { HHTT } \ \text { HTHH } & \text { HTHT } & \text { HTTH } & \text { HTTT } \ \text { THHH } & \text { THHT } & \text { THTH } & \text { THTT } \ \text { TTHH } & \text { TTHT } & \text { TTTH } & \text { TTTT } \end{array}$
Here, for example, HTTH represents the outcome that the first toss is heads, the next two tosses are tails, and the fourth toss is heads. The events $A$ and $B$ are defined as follows.
$\mathrm { A } =$ event exactly two tails are tossed
$B =$ event the first and last tosses are the same
List the outcomes that comprise the event (A & B).

Accepted Answer

Event A requires exactly two tails, and event B requires the first and last tosses to be the same. Combining these conditions, HTTH and THHT are the only outcomes that satisfy both requirements.

Question 3

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
$\mathrm { A } =$ event the first two tosses are the same
$B =$ event the last two tosses are the same. Are A and B independent events?

Accepted Answer

The events A and B are independent because the outcome of the first two tosses does not affect the outcome of the last two tosses.

Question 4

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

Accepted Answer

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

Question 5

Find the conditional 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 the probability that the person's age of
Retirement was between 50 and 55 given that he or she was an attorney.

Accepted Answer

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

Question 6

Use counting rules to determine the probability.
-A committee of 11 members is voting on a proposal. Each member casts a yea or nay vote. On a random voting basis, what is the probability that the proposal wins by a vote of 8 to 3?

Accepted Answer

The probability is calculated by finding the number of favorable outcomes (8 yea votes) divided by the total possible outcomes. The number of ways to choose 8 yea votes out of 11 is given by the combination formula C(11,8). The total number of voting outcomes is 2^11. Thus, the probability is C(11,8)/2^11 = 165/2048 = 0.08057.

Question 7

Provide an appropriate response.
-A contingency table provides a joint frequency distribution for the popular votes cast in a presidential election by sex and political party. A joint probability distribution corresponding to the contingency table is obtained and can be represented as follows.

The letters a through lare used to represent the probabilities in the different cells so, for example, the letter $\mathrm { f }$ represents $\mathrm { P } \left( \mathrm { P } _ { 2 } \& \mathrm {~S} _ { 2 } \right)$ and the letter h represents $\mathrm { P } \left( \mathrm { S } _ { 2 } \right)$.
if a person who voted in this election is selected at random, the events $\mathrm { P } _ { 2 }$ and $\mathrm { S } _ { 2 }$ are mutually exclusive?

Accepted Answer

The answer of Provide an appropriate response.
-A contingency table provides...

Question 8

List the outcomes comprising the specified event.
-When a quarter is tossed four times, 16 outcomes are possible. $\begin{array} { l l l l } \text { HHHH } & \text { HHHT } & \text { HHTH } & \text { HHTT } \ \text { HTHH } & \text { HTHT } & \text { HTTH } & \text { HTTT } \ \text { THHH } & \text { THHT } & \text { THTH } & \text { THTT } \ \text { TTHH } & \text { TTHT } & \text { TTTH } & \text { TTTT } \end{array}$
Here, for example, HTTH represents the outcome that the first toss is heads, the next two tosses are tails, and the fourth toss is heads. List the outcomes that comprise the following event.
$\mathrm { A } =$ event exactly three tails are tossed

Accepted Answer

Event A, which is exactly three tails being tossed, includes the outcomes HTTT, THTT, TTHT, and TTTH. Each of these outcomes has exactly three tails and one head, matching the criteria for event A.

Question 9

Use counting rules to determine the probability.
-Dave puts a collection of 15 books on a bookshelf in a random order. Among the books are 2 fiction and 13 nonfiction books. What is the probability that the 2 fiction books will be all together on the Left side of the shelf and the 13 nonfiction all together on the right side of the shelf?

Accepted Answer

The probability is calculated by considering the arrangement where the 2 fiction books are together on the left and the 13 nonfiction books are on the right. There are 2! ways to arrange the fiction books among themselves and 13! ways to arrange the nonfiction books. Since there are 15! total ways to arrange all books without any restrictions, the probability is (2! * 13!) / 15!, which simplifies to 1/105 or approximately 0.00952.

Question 10

Solve the problem.
-A poker hand consists of 5 cards dealt from an ordinary deck of 52 playing cards. How many different hands are there consisting of four hearts and one spade?

Accepted Answer

The answer of Solve the problem.
-A poker hand consists of...

Question 11

Use the special multiplication rule to find the indicated probability.
-A family has five children. The probability of having a girl is 1/2. What is the probability of having 3 girls followed by 2 boys?

Accepted Answer

The probability of having 3 girls followed by 2 boys is (1/2)^3 * (1/2)^2 = 1/32 = 0.0313.

Question 12

Determine whether the events are independent.
-An auto insurance company was interested in investigating accident rates for drivers in different age groups. The following contingency table was based on a random sample of drivers and Classifies drivers by age group and number of accidents in the past three years.

Suppose that one of the drivers is selected at random. Are the events $\mathrm { G } _ { 1 }$ and $\mathrm { A } _ { 3 }$ independent?

Accepted Answer

The answer of Determine whether the events are independent.
-An auto...

Question 13

Find the indicated probability by using the general addition rule.
-A spinner has regions numbered 1 through 21 . What is the probability that the spinner will stop on an even number or a multiple of 3 ?

Accepted Answer

The even numbers between 1 and 21 are 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20, making 10 even numbers. The multiples of 3 between 1 and 21 are 3, 6, 9, 12, 15, 18, and 21, making 7 multiples of 3. However, 6, 12, and 18 are both even and multiples of 3, so they are counted twice if we simply add the two sets. To avoid double-counting, we subtract the numbers that are both even and multiples of 3, which are 3 numbers. So, we have 10 (even numbers) + 7 (multiples of 3) - 3 (duplicates) = 14 unique numbers that are either even or multiples of 3 out of 21. Therefore, the probability is $ \frac{14}{21} = \frac{2}{3} $.

Question 14

Use the rule of total probability to find the indicated probability.
-Among students at one college are 3874 women and 3058 men. The following table provides relative-frequency distributions for subject major for males and females at the college. $$\begin{array} { l c c } 
\text { Major } & \begin{array} { l } 
\text { Relative frequency } \
\text { for women }
\end{array} & \begin{array} { l } 
\text { Relative frequency } \
\text { for men }
\end{array} \
\hline \text { Humanities } & 0.187 & 0.165 \
\text { Science } & 0.284 & 0.349 \
\text { Social Science } & 0.142 & 0.179 \
\text { Other } & 0.387 & 0.307
\end{array}$$ A student is selected at random from the college. Determine the probability that the student is a Science major.

Accepted Answer

The probability that a randomly selected student is a Science major can be calculated using the rule of total probability. This involves taking the weighted average of the probabilities of being a Science major among women and men, weighted by the proportion of women and men in the college. The calculation is as follows:$$P(\text{Science}) = P(\text{Science}|\text{Woman})P(\text{Woman}) + P(\text{Science}|\text{Man})P(\text{Man})$$Given:- Total students = 3874 women + 3058 men = 6932- $P(\text{Science}|\text{Woman}) = 0.284$- $P(\text{Science}|\text{Man}) = 0.349$- $P(\text{Woman}) = \frac{3874}{6932}$- $P(\text{Man}) = \frac{3058}{6932}$$$P(\text{Science}) = 0.284 \times \frac{3874}{6932} + 0.349 \times \frac{3058}{6932} \approx 0.313$$Therefore, the probability that a randomly selected student is a Science major is approximately 0.313.

Question 15

Provide an appropriate response.
-A contingency table provides a joint frequency distribution for the popular votes cast in a presidential election by sex and political party. A joint probability distribution corresponding to the contingency table is obtained and can be represented as follows.

The letters a through lare used to represent the probabilities in the different cells so, for example, the letter $f$ represents $P \left( P _ { 2 } \& S _ { 2 } \right)$ and the letter $h$ represents $P \left( S _ { 2 } \right)$.
if a person who voted in this election is selected at random, the events $\left( \mathrm { P } _ { 1 } \right.$ \& and $\left( P _ { 2 } \& S _ { 1 } \right)$ are mutually exclusive?

Accepted Answer

The answer of Provide an appropriate response.
-A contingency table provides...

Question 16

Find the indicated probability.
-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 people 16 years old and over.  Suppose one of these people is selected at random. Compute the probability that the person is a
Nonreader.

Accepted Answer

The answer of Find the indicated probability.
-The Book Industry Study...

Question 17

Use the special multiplication rule to find the indicated probability.
-In one large city, 42% of all voters are Democrats. If two voters are randomly selected for a survey, find the probability that they are both Democrats.

Accepted Answer

The probability that both voters are Democrats is found by multiplying the probability of selecting a Democrat twice (assuming independent events): 0.42 * 0.42 = 0.1764, which rounds to 0.176.

Question 18

Use counting rules to determine the probability.
-A student takes a true-false test consisting of 12 questions. Assuming that the student guesses at each question, find the probability that the student answers exactly 10 questions correctly.

Accepted Answer

The probability of guessing exactly 10 out of 12 questions correctly on a true-false test can be calculated using the binomial probability formula, which is P(X=k) = (nCk) * p^k * (1-p)^(n-k), where n is the total number of questions, k is the number of correct answers, p is the probability of getting a question right, and (nCk) is the combination of n items taken k at a time. Here, n=12, k=10, and p=0.5 (since it's a true/false test). Thus, P(10 correct out of 12) = (12C10) * (0.5)^10 * (0.5)^2 = 66 * (0.5)^12 = 0.0161.

Question 19

Use the special multiplication rule to find the indicated probability.
-In one large city, 63% of adults have health insurance. What is the probability that 4 adults selected at random from the town all have health insurance?

Accepted Answer

The probability that all 4 adults selected at random have health insurance is found by multiplying the probability of one having health insurance (0.63) by itself four times (since the events are independent): $0.63^4 = 0.158$.

Question 20

Determine whether the events are independent.
-When a balanced die is rolled twice, 36 equally likely outcomes are possible. Let $\mathrm { A } =$ event the sum of the two rolls is 8 $B =$ event the first roll comes up 3 . Are A and B independent events?

Accepted Answer

The probability of event A occurring given that event B has occurred is different from the probability of event A occurring independently, indicating that A and B are not independent events.

Quiz 4: Probability Concepts

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

Use counting rules to determine the probability. -A committee of 11 members is voting on a proposal. Each member casts a yea or nay vote. On a random voting basis, what is the probability that the proposal wins by a vote of 8 to 3?

Solve the problem. -A poker hand consists of 5 cards dealt from an ordinary deck of 52 playing cards. How many different hands are there consisting of four hearts and one spade?

Use the special multiplication rule to find the indicated probability. -A family has five children. The probability of having a girl is 1/2. What is the probability of having 3 girls followed by 2 boys?

Find the indicated probability by using the general addition rule. -A spinner has regions numbered 1 through 21 . What is the probability that the spinner will stop on an even number or a multiple of 3 ?

Use the special multiplication rule to find the indicated probability. -In one large city, 42% of all voters are Democrats. If two voters are randomly selected for a survey, find the probability that they are both Democrats.

Use counting rules to determine the probability. -A student takes a true-false test consisting of 12 questions. Assuming that the student guesses at each question, find the probability that the student answers exactly 10 questions correctly.

Use the special multiplication rule to find the indicated probability. -In one large city, 63% of adults have health insurance. What is the probability that 4 adults selected at random from the town all have health insurance?

Determine whether the events are independent. -When a balanced die is rolled twice, 36 equally likely outcomes are possible. Let A=\mathrm { A } =A= event the sum of the two rolls is 8 B=B =B= event the first roll comes up 3 . Are A and B independent events?

Determine whether the events are independent. -When a balanced die is rolled twice, 36 equally likely outcomes are possible. Let $\mathrm { A } =$ event the sum of the two rolls is 8 $B =$ event the first roll comes up 3 . Are A and B independent events?