Question 1

Find the expected value of the random variable. Round to the nearest cent unless stated otherwise.
-Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $4.00 for rolling a 4 or a 5, nothing otherwise. What is your expected value? 
A)-$2.00
B)$2.00
C)-$0.67
D)$4.00

Accepted Answer

The expected value is calculated by multiplying each outcome by its probability and summing these products. Here, the probability of rolling a 4 or a 5 is 2/6 (since there are two favorable outcomes out of six possible outcomes), and the payout for these outcomes is $4.00. For any other roll, the outcome is -$2.00 (the cost to play). Thus, the expected value is (2/6)*$4.00 + (4/6)*(-$2.00) = $8/6 - $8/6 = -$0.67.

Question 2

Obtain the probability distribution of the random variable.
-The following table displays a frequency distribution for the number of siblings for students at one middle school. For a randomly selected student in the school, let X denote the number of siblings of
The student. Obtain the probability distribution of X. $\begin{array}{r|rrrrrrrr}
\text { Number of siblings } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \
\hline \text { Frequency } & 192 & 246 & 123 & 56 & 20 & 9 & 5 & 1
\end{array}$
 
A)
$\begin{array}{l}
\text { Siblings Probability }\
\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 1 & 0.535 \
2 & 0.267 \
3 & 0.122 \
4 & 0.043 \
5 & 0.020 \
6 & 0.011 \
7 & 0.002
\end{array}
\end{array}$

B)
$\begin{array}{l}
\text { Siblings Probability }\
\begin{array}{r|r}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.294 \
1 & 0.377 \
2 & 0.189 \
3 & 0.086 \
4 & 0.031 \
5 & 0.014 \
6 & 0.008 \
7 & 0.002
\end{array}
\end{array}$

C)
$\begin{array}{r|r}
\text { Siblings } & \text { Probability } \
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.310 \
1 & 0.362 \
2 & 0.201 \
3 & 0.074 \
4 & 0.031 \
5 & 0.014 \
6 & 0.008 \
7 & 0.002
\end{array}$

D)
$\begin{array}{r|r}
\text { Siblings } & \text { Probability } \
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.125 \
1 & 0.125 \
2 & 0.125 \
3 & 0.125 \
4 & 0.125 \
5 & 0.125 \
6 & 0.125 \
7 & 0.125
\end{array}$

Accepted Answer

To find the probability distribution, divide the frequency of each number of siblings by the total number of students. The total number of students is $192 + 246 + 123 + 56 + 20 + 9 + 5 + 1 = 652$. Calculating the probability for each number of siblings using this total gives the probabilities listed in option B.

Question 3

Calculate the specified probability
-Suppose that $W$ is a random variable. Given that $P ( W \leq 3 ) = 0.1$, find $P ( W > 3 )$.  
A)0
B) 3
C) $0.1$
D) $0.9$

Accepted Answer

The probability that $W$ is greater than 3 is the complement of the probability that $W$ is less than or equal to 3. Since $P(W \leq 3) = 0.1$, $P(W > 3) = 1 - P(W \leq 3) = 1 - 0.1 = 0.9$.

Question 4

Find the standard deviation of the random variable.
-The random variable X is the number of houses sold by a realtor in a single month at the Sendsom's Real Estate office. Its probability distribution is given in the table. Round the answer to two decimal
Places. $\begin{array}{c|cccccccc}
\text { Houses Sold (x) } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \
\hline \text { Probability } \mathrm{P}(\mathrm{x}) & 0.24 & 0.01 & 0.12 & 0.16 & 0.01 & 0.14 & 0.11 & 0.21
\end{array}$
A)$6.86$
B) $4.45$
C) $2.62$
D) $2.25$

Accepted Answer

To find the standard deviation of a discrete random variable, we first need to calculate its mean ($\mu$) and then use the formula for the standard deviation ($\sigma$) which involves summing the squared differences between each value and the mean, weighted by their probabilities. 1. Calculate the mean ($\mu$):$\mu = \sum xP(x) = (0)(0.24) + (1)(0.01) + (2)(0.12) + (3)(0.16) + (4)(0.01) + (5)(0.14) + (6)(0.11) + (7)(0.21) = 3.59$2. Calculate the variance ($\sigma^2$):$\sigma^2 = \sum (x - \mu)^2P(x)$$\sigma^2 = (0-3.59)^2(0.24) + (1-3.59)^2(0.01) + (2-3.59)^2(0.12) + (3-3.59)^2(0.16) + (4-3.59)^2(0.01) + (5-3.59)^2(0.14) + (6-3.59)^2(0.11) + (7-3.59)^2(0.21)$$\sigma^2 = 6.8641$3. Calculate the standard deviation ($\sigma$):$\sigma = \sqrt{\sigma^2} = \sqrt{6.8641} \approx 2.62$Therefore, the standard deviation of the random variable is approximately 2.62.

Question 5

Find the mean of the random variable.
-The random variable X is the number that shows up when a loaded die is rolled. Its probability distribution is given in the table. Round the answer to two decimal places. $\begin{array}{c|cccccc}
\mathrm{X} & 1 & 2 & 3 & 4 & 5 & 6 \
\hline \mathrm{P}(\mathrm{X}=\mathrm{x}) & 0.16 & 0.11 & 0.13 & 0.13 & 0.10 & 0.37
\end{array}$
A)$0.17$
B) $3.50$
C) $3.88$
D) $4.01$

Accepted Answer

The answer of Find the mean of the random variable.
-The...

Question 6

Find the indicated probability. Round to four decimal places.
-A test consists of 10 true/false questions. To pass the test a student must answer at least 9 questions correctly. If a student guesses on each question, what is the probability that the student will pass
The test? 
A)0.9990
B)0.0010
C)0.0102
D)0.0107

Accepted Answer

The answer of Find the indicated probability. Round to four...

Question 7

Find the expected value of the random variable. Round to the nearest cent unless stated otherwise.
-In a game, you have a 1/33 probability of winning $62 and a 32/33 probability of losing $2. What is your expected value? 
A)-$1.94
B)$1.88
C)$3.82
D)-$0.06

Accepted Answer

The expected value (EV) is calculated by multiplying each outcome by its probability and summing these products. EV = (1/33 * $62) + (32/33 * -$2) = $62/33 - $64/33 = -$2/33, which is approximately -$0.06.

Question 8

Construct a probability histogram for the binomial random variable, X.
-Three coins are tossed. X is the number of tails. 
A)

B)

C)

D)

Accepted Answer

The answer of Construct a probability histogram for the binomial...

Question 9

Let the random variable X represent the winnings at one play of a particular game. The expected value of X is known to be -$0.32. This means that on any given play of the game, the most likely outcome is that the player will lose 32 cents?

Accepted Answer

The expected value of -$0.32 means that on average, over many plays, a player will lose 32 cents per play. It does not necessarily indicate the most likely outcome of a single play.

Question 10

Find the specified probability.
-Use the special addition rule and the following probability distribution to determine P(X = 6). $\begin{array}{cccccccc}
\mathrm{X} & 5 & 6 & 7 & 8 & 9 & 10 & 11 \
\hline \mathrm{P}(\mathrm{X}=\mathrm{x}) & 0.05 & 0.05 & 0.20 & 0.15 & 0.15 & 0.10 & 0.30
\end{array}$
A)$0.10$
B) $0.95$
C) $0.90$
D) $0.05$

Accepted Answer

The probability P(X = 6) is directly given in the table as 0.05.

Question 11

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary.
-The number of calls received by a mountain search and rescue team in a day has a Poisson distribution with parameter $\lambda$ = 0.70. Find the probability that on a randomly selected day, they
Will receive fewer than two calls. 
A)0.156
B)0.348
C)0.122
D)0.844

Accepted Answer

The probability of receiving fewer than two calls (0 or 1 call) with $\lambda = 0.70$ is calculated using the Poisson distribution formula: $P(X < 2) = P(0) + P(1)$. For $P(0)$, the formula is $e^{-\lambda} \lambda^0 / 0! = e^{-0.70}$. For $P(1)$, it's $e^{-\lambda} \lambda^1 / 1! = 0.70e^{-0.70}$. Adding these gives $e^{-0.70} + 0.70e^{-0.70}$, which equals approximately 0.844.

Question 12

Determine the possible values of the random variable.
-For a randomly selected student in a particular high school, let Y denote the number of living grandparents of the student. What are the possible values of the random variable Y? 
A)1, 2, 3, 4
B)4
C)0, 1, 2
D)0, 1, 2, 3, 4

Accepted Answer

The number of living grandparents a student can have ranges from 0 (if all have passed away) to 4 (if all are still living), including all whole numbers in between.

Question 13

Provide an appropriate response.
-Which of the random variables described below is likely to have a Poisson distribution? - The random variable X is the number of accidents occurring during a ski season at a particular
Ski resort.-Fifty people are selected at random from among the skiers at a particular resort. The random Variable Y is the number among the fifty who have been involved in a ski accident during the past Ski season. -One ski resort allows both skiers and snowboarders. Fifty people are selected at random from the People waiting in line at one of the resort's chairlifts. The random variable Z is the number of Snowboarders among the fifty. 
A)X, Y, and Z
B)Y and Z
C)X only
D)X and Y

Accepted Answer

The Poisson distribution is typically used to model the number of times an event happens in a fixed interval of time or space under certain conditions, such as the number of accidents occurring during a ski season at a particular resort (X). Y and Z are not Poisson distributions because they involve selecting a fixed number of individuals and counting occurrences among them, which is more suited to binomial distribution scenarios.

Question 14

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary.
-The number of power failures experienced by the Columbia Power Company in a day has a Poisson distribution with parameter $\lambda$ = 0.210. Find the probability that there are exactly two
Power failures in a particular day. 
A)0.027
B)0.085
C)0.036
D)0.018

Accepted Answer

The answer of Use the Poisson Distribution to find the...

Question 15

Determine the required probability by using the Poisson approximation to the binomial distribution. Round to threedecimal places.
-The rate of defects among CD players of a certain brand is 1.6%. Use the Poisson approximation to the binomial distribution to find the probability that among 450 such CD players received by a Store, there are exactly three defectives. 
A)0.070
B)0.093
C)0.035
D)0.046

Accepted Answer

The answer of Determine the required probability by using the...

Question 16

Calculate the specified probability
-Suppose that $D$ is a random variable. Given that $P ( D > 1.4 ) = 0.95$, find $P ( D \leq 1.4 )$.  
A)$0.95$
B) $0.05$
C) $0.025$
D) 0

Accepted Answer

The probability that $D \leq 1.4$ is the complement of $D > 1.4$, which is $1 - 0.95 = 0.05$.

Question 17

Find the indicated probability. Round to four decimal places.
-The participants in a television quiz show are picked from a large pool of applicants with approximately equal numbers of men and women. Among the last 10 participants there have been
Only 2 women. If participants are picked randomly, what is the probability of getting 2 or fewer Women when 10 people are picked? 
A)0.0547
B)0.0439
C)0.0107
D)0.0537

Accepted Answer

This is a binomial probability problem where we are looking for the probability of getting 2 or fewer successes (women in this case) out of 10 trials (participants), with the probability of success on a single trial being 0.5 (since there are equal numbers of men and women). The formula for binomial probability is P(X = k) = (nCk) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success on a single trial, and nCk is the binomial coefficient. To find the probability of getting 2 or fewer women, we sum the probabilities of getting exactly 0, 1, and 2 women. Calculating these probabilities and summing them gives a result that rounds to 0.0547, making option A correct.

Question 18

Find the mean of the binomial random variable. Round to two decimal places when necessary.
-The probability is 0.3 that a person shopping at a certain store will spend less than $20. For groups of size 16, find the mean number who spend less than $20. 
A)14
B)11.2
C)6
D)4.8

Accepted Answer

The mean of a binomial random variable is given by the formula μ = np, where n is the number of trials (in this case, the group size of 16) and p is the probability of success (in this case, 0.3). Thus, the mean is 16 * 0.3 = 4.8.

Question 19

Calculate the specified probability
-Suppose that $\mathrm { T }$ is a random variable. Given that $\mathrm { P } ( 3.5 \leq \mathrm { T } \leq 3.5 ) = 0.275$, and that $\mathrm { P } ( \mathrm { K } < 3.5 ) = \mathrm { P } ( \mathrm { K } >$ 3.5), find $\mathrm { P } ( \mathrm { K } < - 3.5 )$.  
A)$0.3625$
B) $0.275$
C) $1.75$
D) $0.725$

Accepted Answer

Given that $P(3.5 \leq T \leq 3.5) = 0.275$, this implies that the probability of $T$ being exactly 3.5 is 0.275. Since $P(K < 3.5) = P(K > 3.5)$, the total probability for $K$ being less than or greater than 3.5 must sum to $1 - 0.275 = 0.725$, because probabilities sum to 1 and we subtract the probability of $K$ being exactly 3.5. Given the symmetry, $P(K < 3.5) = P(K > 3.5) = 0.725 / 2 = 0.3625$. Since $P(K < -3.5)$ is a subset of $P(K < 3.5)$ and the distribution is not specified, we assume the distribution is symmetric around 0, making $P(K < -3.5) = P(K > 3.5) = 0.3625$.

Question 20

Provide an appropriate response.
-Three random variables X, Y, and Z, are described below. In which of these situations would it be acceptable to use the binomial distribution?
A: A bag contains 4 blue marbles and 8 red marbles. Five marbles are drawn at random with
Replacement. The random variable X is the number of blue marbles drawn.
B: A bag contains 4 blue marbles and 8 red marbles. Six marbles are drawn at random without
Replacement. The random variable Y is the number of blue marbles drawn.
C: A bag contains 30 blue marbles and 38 red marbles. Three marbles are drawn at random without Replacement. The random variable Z is the number of blue marbles drawn. 
A)A only
B)B and C
C)A, B, and C
D)A and C

Accepted Answer

The binomial distribution is appropriate when each trial is independent and the probability of success is constant. Choice A involves drawing with replacement, which satisfies these conditions. Choices B and C involve drawing without replacement, which means the trials are not independent and the probability of success changes with each draw, making the binomial distribution inappropriate.

Quiz 5: Discrete Random Variables

Find the expected value of the random variable. Round to the nearest cent unless stated otherwise. -Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $4.00 for rolling a 4 or a 5, nothing otherwise. What is your expected value?

Calculate the specified probability -Suppose that $W$ is a random variable. Given that $P ( W \leq 3 ) = 0.1$ , find $P ( W > 3 )$ .

Find the indicated probability. Round to four decimal places. -A test consists of 10 true/false questions. To pass the test a student must answer at least 9 questions correctly. If a student guesses on each question, what is the probability that the student will pass The test?

Find the expected value of the random variable. Round to the nearest cent unless stated otherwise. -In a game, you have a 1/33 probability of winning $62 and a 32/33 probability of losing $2. What is your expected value?

Construct a probability histogram for the binomial random variable, X. -Three coins are tossed. X is the number of tails.

Let the random variable X represent the winnings at one play of a particular game. The expected value of X is known to be -$0.32. This means that on any given play of the game, the most likely outcome is that the player will lose 32 cents?

Find the specified probability. -Use the special addition rule and the following probability distribution to determine P(X = 6) . $\begin{array}{cccccccc}\mathrm{X} & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\\hline \mathrm{P}(\mathrm{X}=\mathrm{x}) & 0.05 & 0.05 & 0.20 & 0.15 & 0.15 & 0.10 & 0.30\end{array}$

Determine the possible values of the random variable. -For a randomly selected student in a particular high school, let Y denote the number of living grandparents of the student. What are the possible values of the random variable Y?

Calculate the specified probability -Suppose that $D$ is a random variable. Given that $P ( D > 1.4 ) = 0.95$ , find $P ( D \leq 1.4 )$ .

Find the mean of the binomial random variable. Round to two decimal places when necessary. -The probability is 0.3 that a person shopping at a certain store will spend less than $20. For groups of size 16, find the mean number who spend less than $20.

Calculate the specified probability -Suppose that $\mathrm { T }$ is a random variable. Given that $\mathrm { P } ( 3.5 \leq \mathrm { T } \leq 3.5 ) = 0.275$ , and that $\mathrm { P } ( \mathrm { K } < 3.5 ) = \mathrm { P } ( \mathrm { K } >$ 3.5) , find $\mathrm { P } ( \mathrm { K } < - 3.5 )$ .

Quiz 5: Discrete Random Variables

Find the expected value of the random variable. Round to the nearest cent unless stated otherwise. -Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $4.00 for rolling a 4 or a 5, nothing otherwise. What is your expected value?

Calculate the specified probability -Suppose that WWW is a random variable. Given that P(W≤3) =0.1P ( W \leq 3 ) = 0.1P(W≤3) =0.1 , find P(W>3) P ( W > 3 ) P(W>3) .

Find the indicated probability. Round to four decimal places. -A test consists of 10 true/false questions. To pass the test a student must answer at least 9 questions correctly. If a student guesses on each question, what is the probability that the student will pass The test?

Find the expected value of the random variable. Round to the nearest cent unless stated otherwise. -In a game, you have a 1/33 probability of winning $62 and a 32/33 probability of losing $2. What is your expected value?

Construct a probability histogram for the binomial random variable, X. -Three coins are tossed. X is the number of tails.

Let the random variable X represent the winnings at one play of a particular game. The expected value of X is known to be -$0.32. This means that on any given play of the game, the most likely outcome is that the player will lose 32 cents?

Determine the possible values of the random variable. -For a randomly selected student in a particular high school, let Y denote the number of living grandparents of the student. What are the possible values of the random variable Y?

Calculate the specified probability -Suppose that DDD is a random variable. Given that P(D>1.4) =0.95P ( D > 1.4 ) = 0.95P(D>1.4) =0.95 , find P(D≤1.4) P ( D \leq 1.4 ) P(D≤1.4) .

Find the mean of the binomial random variable. Round to two decimal places when necessary. -The probability is 0.3 that a person shopping at a certain store will spend less than $20. For groups of size 16, find the mean number who spend less than $20.

Calculate the specified probability -Suppose that $W$ is a random variable. Given that $P ( W \leq 3 ) = 0.1$ , find $P ( W > 3 )$ .

Calculate the specified probability -Suppose that $D$ is a random variable. Given that $P ( D > 1.4 ) = 0.95$ , find $P ( D \leq 1.4 )$ .