Question 1

Give the probability distribution for the indicated random variable.  
A fair coin is tossed three times, and we count the number of "runs" that occur. A "run" is defined as a sequence of one or more of the same letter (e.g., HTH is 3 runs, TTH is 2 runs, and TTT is 1 run).
 
A) $\begin{array} { l } 
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } { 6 } &\frac { 2 } { 3 } & \frac{ 1 } { 6 }  \
\end{array}$  
B)   $\begin{array} { l } 
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } { 3 } &\frac { 1 } { 3 } & \frac{ 1 } {3 }  \
\end{array}$
 
C)   $\begin{array} { l } 
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } { 9 } &\frac { 7 } { 9 } & \frac{ 1 } { 9 }  \
\end{array}$
 
D)   $\begin{array} { l } 
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } {8 } &\frac { 3 } { 4 } & \frac{ 1 } { 8 }  \
\end{array}$
 
E)   $\begin{array} { l } 
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } { 4 } &\frac { 1 } { 2 } & \frac{ 1 } { 4 }  \
\end{array}$

Accepted Answer

There are 8 possible outcomes when a fair coin is tossed three times: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. The number of runs for each outcome is: 1 run (HHH, TTT), 2 runs (HHT, HTT, THH, TTH), and 3 runs (HTH, THT). The probabilities are calculated as follows: P(X=1) = 2/8 = 1/4, P(X=2) = 4/8 = 1/2, P(X=3) = 2/8 = 1/4.

Question 2

According to a study, the probability that a randomly selected teenager watched a rented video at least once during a week was 0.76. What is the probability that at least 8 teenagers in a group of 10 watched a rented movie at least once last week  Round your answer to two decimal places.

Accepted Answer

The answer of According to a study, the probability that...

Question 3

The voltage of an electrical outlet in your home is an example of a discrete random variable.

Accepted Answer

The voltage of an electrical outlet typically follows a continuous distribution, not a discrete distribution.

Question 4

The probability that a randomly selected pound of beef is purchased by McDonald's is 0.06. Ten pounds of beef are chosen at random.
What is the probability that exactly two of them are purchased by McDonald's  Round your answer to four decimal places.
The probability is __________.
Use technology to generate the probability distribution for the associated binomial random variable. Round your answers to four decimal places. $\begin{array} { | c | c | c | c | c | c | } 
\hline x & 0 & 1 & 2 & 3 & 4 \
\hline P ( X = x ) & & & & & \
\hline 5 & 6 & 7 & 8 & 9 & 10 \
\hline & & & & & \
\hline
\end{array}$ ?
If 10 pounds of beef produced in the U.S. are selected at random, the number of pounds most likely to have been purchased by McDonald's is __________.

Accepted Answer

The answer of The probability that a randomly selected pound...

Question 5

A random variable has the probability distribution table as shown. Calculate $$P ( x \leq - 1 )$$ . $$\begin{array} { | l | l | l | l | l | l | } 
\hline \boldsymbol { x } & - 3 & - 2 & - 1 & 0 & 1 \
\hline \boldsymbol { P } ( \boldsymbol { X } = \boldsymbol { x } ) & & & 0.5 & 0.1 & 0.1 \
\hline
\end{array}$$  
A)0.5
B) 0.3
C) 0.8
D) 0
E) Cannot be determined from given information.

Accepted Answer

The answer of A random variable has the probability distribution...

Question 6

Give the probability distribution for the indicated random variable.  
A black and white die are rolled, and consider the following random variable.
  $X = \left\{ \begin{array} { l } 
0 , \text { if both numbers are odd } \
3 , \text { if one number is odd and one is even } \
6 , \text { if both numbers are even }
\end{array} \right.$  
A)  $\begin{array} { l } 
x & 0 & 3& 6 \
P ( X = x ) &\frac { 1 } { 4 } &\frac { 1 } { 4 } & \frac{ 1 } { 2 }  \
\end{array}$  
B)   $\begin{array} { l } 
x & 0 & 3& 6 \
P ( X = x ) &\frac { 2 } { 3} &\frac { 1 } { 6 } & \frac{ 1 } { 6 }  \
\end{array}$
 
C) $\begin{array} { l } 
x & 0 & 3& 6 \
P ( X = x ) &\frac { 1 } { 3 } &\frac { 1 } { 3 } & \frac{ 1 } { 3 }  \
\end{array}$  
D)   $\begin{array} { l } 
x & 0 & 3& 6 \
P ( X = x ) &\frac { 1 } { 2 } &\frac { 1 } { 4 } & \frac{ 1 } { 4 }  \
\end{array}$
 
E)   $\begin{array} { l } 
x & 0 & 3& 6 \
P ( X = x ) &\frac { 1 } { 4 } &\frac { 1 } {2 } & \frac{ 1 } { 4 }  \
\end{array}$

Accepted Answer

Each die has 3 odd (1, 3, 5) and 3 even (2, 4, 6) numbers. The probability of both odd is $(\frac{3}{6})^2 = \frac{1}{4}$, both even is also $(\frac{3}{6})^2 = \frac{1}{4}$, and one odd one even (or vice versa) is $2 \times \frac{3}{6} \times \frac{3}{6} = \frac{1}{2}$.

Question 7

In a large on-the-job training program, half of the participants are female and half are male. In a random sample of 7 participants, what is the probability that an investigator will draw at least 1 female  Please round your answer to four decimal places.

Accepted Answer

The answer of In a large on-the-job training program, half...

Question 8

The mathematics final exam scores for the students in your study group are 89%, 83%, 95%, 67%, 92%, and 80%. ​
List the values of X for all the outcomes.
​
A)89%, 83%, 95%, 67%, 92%
B) 89%, 83%, 67%, 92%, 80%
C) 89%, 83%, 95%, 67%
D) 89%, 83%, 95%, 67%, 92%, 80%
E) 11%, 17%, 5%, 33%, 8%, 20%

Accepted Answer

The given scores are 89%, 83%, 95%, 67%, 92%, and 80%, which matches only with option D. Option A and C have fewer scores than provided, while option B has a different set of scores. Option E has completely different values of X, which are not related to the given problem. Therefore, option D is the best choice.

Question 9

The following table shows the average percentage increase in the price of a house from 1980 to 2001 in 9 regions of the U.S. $\begin{array} { | l | l | } 
\hline \text { New England } & 300 \
\hline \text { Pacific } & 225 \
\hline \text { Middle Atlantic } & 225 \
\hline \text { South Atlantic } & 175 \
\hline \text { Mountain } & 175 \
\hline \text { West North Central } & 100 \
\hline \text { West South Central } & 75 \
\hline \text { East North Central } & 175 \
\hline \text { East South Central } & 100 \
\hline
\end{array}$

Let  X be the percentage increase in the price of a house in a randomly selected region.
What is the probability that, in a randomly selected region, the percentage increase in the cost of a house exceeded 200% 
 
A) $P ( X > 200 ) = \frac { 1 } { 9 }$
B)   $P ( X > 200 ) = \frac { 2 } { 3 }$
C)   $P ( X > 200 ) = \frac { 1 } { 3 }$
D)   $P ( X > 200 ) = \frac { 1 } { 2 }$
E)   $P ( X > 200 ) = \frac { 2 } { 9 }$

Accepted Answer

The answer of The following table shows the average percentage...

Question 10

The probability that a randomly chosen citizen-entity of Cygnus is of pension age is approximately 0.7. What is the probability that, in a randomly selected sample of 3 citizen-entities, all of them are of pension age  Round your answer to four decimal places.

Accepted Answer

The answer of The probability that a randomly chosen citizen-entity...

Question 11

In a large on-the-job training program, half of the participants are female and half are male. In a random sample of 5 participants, what is the probability that an investigator will draw at least 2 females  Round your answer to four decimal places.

Accepted Answer

The answer of In a large on-the-job training program, half...

Question 12

Your scores for the 20 surprise math quizzes last semester were (out of 10)   $\begin{array} { l l l l l l l l l l l } 
5.5&9.5&10.0&2.5&8.0&9.5&7.5&6.5&7.0&8.0\
8.0&8.5&7.5&7.0&8.0&10.0&10.0&8.5&7.5&8.0
\end{array}$  
Use these raw data to construct a frequency table with the following brackets: 2.1 - 4.0, 4.1 - 6.0, 6.1 - 8.0, 8.1 - 10.0. Find the probability distribution using the (rounded) midpoint values as the values of X.
 
A) 
$\begin{array} { l } 
x & 3.0&5.0&7.0&9.0\
P ( X = x ) &0.05&0.55&0.05&0.35 \
\end{array}$

B) 
  $\begin{array} { l } 
x & 3.0&5.0&7.0&9.0\
P ( X = x ) &0.05&0.05&0.55&0.35 \
\end{array}$
 
C)
   $\begin{array} { l } 
x & 3.0&5.0&7.0&9.0\
P ( X = x ) &0.05&0.55&0.35&0.05 \
\end{array}$
 
D)   
$\begin{array} { l } 
x & 3.0&5.0&7.0&9.0\
P ( X = x ) &0.35&0.55&0.05&0.05 \
\end{array}$
 
E)  
 $\begin{array} { l } 
x & 3.0&5.0&7.0&9.0\
P ( X = x ) &0.05&0.05&0.55&0.35 \
\end{array}$.

Accepted Answer

First, we categorize the scores into the given brackets: - 2.1 - 4.0: 1 score (2.5)- 4.1 - 6.0: 1 score (5.5)- 6.1 - 8.0: 11 scores- 8.1 - 10.0: 7 scoresThere are 20 scores in total. The probability for each bracket is calculated by dividing the number of scores in the bracket by the total number of scores:- P(X=3.0) = 1/20 = 0.05- P(X=5.0) = 1/20 = 0.05- P(X=7.0) = 11/20 = 0.55- P(X=9.0) = 7/20 = 0.35

Question 13

A random variable has the probability distribution table as shown. Calculate $P ( x \geq 5 )$ . $\begin{array} { | l | l | l | l | l | l | } 
\hline \boldsymbol { x } & 1 & 3 & 5 & 7 & 9 \
\hline \boldsymbol { P } ( \boldsymbol { X } = \boldsymbol { x } ) & 0.2 & 0.1 & 0.3 & & 0.2 \
\hline
\end{array}$  
A)0.4
B) 0.5
C) 0.2
D) 0.7
E) 0.8

Accepted Answer

To find $P(x \geq 5)$, we sum the probabilities for $x = 5, 7, \text{ and } 9$. Given $P(X=5) = 0.3$ and $P(X=9) = 0.2$, and knowing the total probability must sum to 1, we can calculate $P(X=7)$ as $1 - (0.2 + 0.1 + 0.3 + 0.2) = 0.2$. Therefore, $P(x \geq 5) = 0.3 + 0.2 + 0.2 = 0.7$.

Question 14

Select an electric utility company at random; R is the exact amount of electricity, in gigawatt hours, it supplies in a year. ​
Classify the random variable R as finite, discrete infinite, or continuous.
​
A)R is the finite random variable.
B) R is the continuous random variable.
C) R is the discrete infinite random variable.

Accepted Answer

The random variable R can take on any non-negative value, including fractional values. Since there are infinite possible values for R within any interval, it is a continuous random variable.

Question 15

According to a July, 1999, article in the New York Times, 13.5% of Internet stocks that entered the market in 1999 ended up trading below their initial offering prices. If you were an investor who purchased 3 Internet stocks at their initial offering prices, what was the probability that at least 2 of them would end up trading at or above their initial offering price  (Round your answer to four decimal places.)

Accepted Answer

The answer of According to a July, 1999, article in...

Question 16

Your class is given a mathematics exam worth 150 points; X is the average score, rounded to the nearest whole number. ​
Classify the random variable X as finite, discrete infinite, or continuous.
​
A)continuous
B) discrete infinite
C) finite

Accepted Answer

X is finite because there is a limited number of average scores that can be obtained from a test scored out of 150 points, making the set of possible values countable and limited.

Question 17

A fair die is rolled. Let the random variable X denote the number that falls bottommost on the die. Find the probability distribution of X.  
A) $\begin{array} { l } 
x & 1 & 2 & 3 & 4 & 5 & 6\
P ( X = x )& \frac { 1 } { 12 } & \frac { 1 } { 6 } &\frac { 1 } { 4 }& \frac { 1 } { 4 } &\frac { 1 } { 6} &\frac { 1 } { 12 } \
\end{array}$  
B)   $\begin{array} { l }

x & 1 & 2 & 3 & 4 & 5 & 6\
P ( X = x ) &\frac { 1 } { 12 } &\frac { 1 } { 12 }& \frac { 1 } { 3 } & \frac { 1 } { 6 } & \frac { 1 } { 12 } & \frac { 1 } { 12 } \
\end{array}$
 
C)   $\begin{array} { l }

x & 1 & 2 & 3 & 4 & 5 & 6\
P ( X = x ) &\frac { 1 } { 6 } &\frac { 1 } { 6 }& \frac { 1 } { 6 } & \frac { 1 } { 6 } & \frac { 1 } { 6} & \frac { 1 } { 6 } \
\end{array}$
 
D)   $\begin{array} { l } 
x & 1 & 2 & 3 & 4 & 5 & 6\
P ( X = x ) &\frac { 1 } { 12 } &\frac { 1 } { 12 } & \frac{ 1 } { 12 } &\frac { 1 } { 12 } & \frac { 1 } { 12 } & \frac { 1 } { 12 } \
\end{array}$
 
E)   $\begin{array} { l } 
x & 1 & 2 & 3 & 4 & 5 & 6\
P ( X = x ) &\frac { 1 } { 3 } &\frac { 1 } { 3 } & \frac{ 1 } { 3 } &\frac { 1 } {3} & \frac { 1 } { 3 } & \frac { 1 } { 3 } \
\end{array}$

Accepted Answer

Each face of a fair die has an equal probability of landing bottommost, making the probability for each outcome $1/6$.

Question 18

X is a binomial variable with $n = 6$ and $p = 0.4$ . Compute $P ( X \leq 2 )$ . Round your answer to five decimal places.
​ $P ( X \leq 2 ) =$ __________

Accepted Answer

The answer of X is a binomial variable with \(n...

Question 19

​Assume that on a standardized test of 100 questions, a person has a probability of 84% of answering any particular question correctly. Find the probability of answering between 75 and 85 questions, inclusive, correctly. (Assume independence, and round your answer to four decimal places.)
​

Accepted Answer

The answer of ​Assume that on a standardized test of...

Question 20

X is a binomial variable with $n = 6$ and $p = 0.5$ . Compute $P ( X = 4 )$ . Round your answer to five decimal places.
​ $P ( X = 4 ) =$ __________

Accepted Answer

The answer of X is a binomial variable with \(n...

Quiz 9: Random Variables and Statistics

Give the probability distribution for the indicated random variable. A fair coin is tossed three times, and we count the number of "runs" that occur. A "run" is defined as a sequence of one or more of the same letter (e.g., HTH is 3 runs, TTH is 2 runs, and TTT is 1 run) .

According to a study, the probability that a randomly selected teenager watched a rented video at least once during a week was 0.76. What is the probability that at least 8 teenagers in a group of 10 watched a rented movie at least once last week Round your answer to two decimal places.

The voltage of an electrical outlet in your home is an example of a discrete random variable.

In a large on-the-job training program, half of the participants are female and half are male. In a random sample of 7 participants, what is the probability that an investigator will draw at least 1 female Please round your answer to four decimal places.

The mathematics final exam scores for the students in your study group are 89%, 83%, 95%, 67%, 92%, and 80%. List the values of X for all the outcomes.

The probability that a randomly chosen citizen-entity of Cygnus is of pension age is approximately 0.7. What is the probability that, in a randomly selected sample of 3 citizen-entities, all of them are of pension age Round your answer to four decimal places.

In a large on-the-job training program, half of the participants are female and half are male. In a random sample of 5 participants, what is the probability that an investigator will draw at least 2 females Round your answer to four decimal places.

Select an electric utility company at random; R is the exact amount of electricity, in gigawatt hours, it supplies in a year. Classify the random variable R as finite, discrete infinite, or continuous.

Your class is given a mathematics exam worth 150 points; X is the average score, rounded to the nearest whole number. Classify the random variable X as finite, discrete infinite, or continuous.

A fair die is rolled. Let the random variable X denote the number that falls bottommost on the die. Find the probability distribution of X.

X is a binomial variable with $n = 6$ and $p = 0.4$ . Compute $P ( X \leq 2 )$ . Round your answer to five decimal places. $P ( X \leq 2 ) =$ __________

Assume that on a standardized test of 100 questions, a person has a probability of 84% of answering any particular question correctly. Find the probability of answering between 75 and 85 questions, inclusive, correctly. (Assume independence, and round your answer to four decimal places.)

X is a binomial variable with $n = 6$ and $p = 0.5$ . Compute $P ( X = 4 )$ . Round your answer to five decimal places. $P ( X = 4 ) =$ __________

Quiz 9: Random Variables and Statistics

Give the probability distribution for the indicated random variable. A fair coin is tossed three times, and we count the number of "runs" that occur. A "run" is defined as a sequence of one or more of the same letter (e.g., HTH is 3 runs, TTH is 2 runs, and TTT is 1 run) .

According to a study, the probability that a randomly selected teenager watched a rented video at least once during a week was 0.76. What is the probability that at least 8 teenagers in a group of 10 watched a rented movie at least once last week Round your answer to two decimal places.

The voltage of an electrical outlet in your home is an example of a discrete random variable.

In a large on-the-job training program, half of the participants are female and half are male. In a random sample of 7 participants, what is the probability that an investigator will draw at least 1 female Please round your answer to four decimal places.

The mathematics final exam scores for the students in your study group are 89%, 83%, 95%, 67%, 92%, and 80%. ​ List the values of X for all the outcomes. ​

The probability that a randomly chosen citizen-entity of Cygnus is of pension age is approximately 0.7. What is the probability that, in a randomly selected sample of 3 citizen-entities, all of them are of pension age Round your answer to four decimal places.

In a large on-the-job training program, half of the participants are female and half are male. In a random sample of 5 participants, what is the probability that an investigator will draw at least 2 females Round your answer to four decimal places.

Select an electric utility company at random; R is the exact amount of electricity, in gigawatt hours, it supplies in a year. ​ Classify the random variable R as finite, discrete infinite, or continuous. ​

Your class is given a mathematics exam worth 150 points; X is the average score, rounded to the nearest whole number. ​ Classify the random variable X as finite, discrete infinite, or continuous. ​

A fair die is rolled. Let the random variable X denote the number that falls bottommost on the die. Find the probability distribution of X.

X is a binomial variable with n=6n = 6n=6 and p=0.4p = 0.4p=0.4 . Compute P(X≤2)P ( X \leq 2 )P(X≤2) . Round your answer to five decimal places. ​ P(X≤2)=P ( X \leq 2 ) =P(X≤2)= __________

​Assume that on a standardized test of 100 questions, a person has a probability of 84% of answering any particular question correctly. Find the probability of answering between 75 and 85 questions, inclusive, correctly. (Assume independence, and round your answer to four decimal places.) ​

X is a binomial variable with n=6n = 6n=6 and p=0.5p = 0.5p=0.5 . Compute P(X=4)P ( X = 4 )P(X=4) . Round your answer to five decimal places. ​ P(X=4)=P ( X = 4 ) =P(X=4)= __________

The mathematics final exam scores for the students in your study group are 89%, 83%, 95%, 67%, 92%, and 80%. List the values of X for all the outcomes.

Select an electric utility company at random; R is the exact amount of electricity, in gigawatt hours, it supplies in a year. Classify the random variable R as finite, discrete infinite, or continuous.

Your class is given a mathematics exam worth 150 points; X is the average score, rounded to the nearest whole number. Classify the random variable X as finite, discrete infinite, or continuous.

X is a binomial variable with $n = 6$ and $p = 0.4$ . Compute $P ( X \leq 2 )$ . Round your answer to five decimal places. $P ( X \leq 2 ) =$ __________

Assume that on a standardized test of 100 questions, a person has a probability of 84% of answering any particular question correctly. Find the probability of answering between 75 and 85 questions, inclusive, correctly. (Assume independence, and round your answer to four decimal places.)

X is a binomial variable with $n = 6$ and $p = 0.5$ . Compute $P ( X = 4 )$ . Round your answer to five decimal places. $P ( X = 4 ) =$ __________