Question 1

An "instant lottery" is played by buying a ticket and scratching off a coating to reveal whether or not you have won a prize, and if so, how much. Suppose an instant lottery pays $5 with probability 0.05 and $100 with probability 0.006. Otherwise it pays nothing. Define 
X = amount won for a single ticket. (You can ignore the cost of the ticket.)
-Let $S be the standard deviation of X. Which of the following is a correct interpretation of this value?
A) We would expect a randomly chosen ticket to result in a win of $D.
B) The most likely value for the amount won for a single ticket is $D.
C) The amount won for a single ticket will deviate, on average, from the expected amount won by about $D.
D) If you were to play the instant lottery many times, the average amount won per ticket would be around $D.

Accepted Answer

The standard deviation measures the spread of the distribution. It tells us how much the amount won for a single ticket will deviate, on average, from the expected amount won.

Question 2

Based on her past experience, a professor knows that the probability distribution function for X = number of students who come to her office hours on any given Wednesday is given below. 
-What is the value of P(X = 4) (the question mark in the table)?
A) 0.05
B) 0.10
C) 0.25
D) 1

Accepted Answer

The answer of Based on her past experience, a professor...

Question 3

Based on her past experience, a professor knows that the probability distribution function for X = number of students who come to her office hours on any given Wednesday is given below. 
-The expected value of X is 1.85. Interpret this value in the context of the problem.
A) If you were to observe many students, the average number of students that visit office hours is 1.85.
B) If you were to observe many Wednesdays, the average number of students that visit office hours is 1.85.
C) On average, the number of students that visit office hours will deviate from the mean by 1.85.
D) Next Wednesday, we expect 2 students to visit office hours.

Accepted Answer

The expected value of X represents the average number of students visiting office hours over many observations, such as many Wednesdays.

Question 4

Based on her past experience, a professor knows that the probability distribution function for X = number of students who come to her office hours on any given Wednesday is given below. 
-The standard deviation of X is 0.96. Interpret this value in the context of the problem.
A) If you were to observe many students, the average number of students that visit office hours is 0.96.
B) If you were to observe many Wednesdays, the average number of students that visit office hours is 0.96.
C) On average, the number of students that visit office hours will deviate from the mean by about one student.
D) Next Wednesday, we expect one student to visit office hours.

Accepted Answer

The standard deviation measures the average deviation of data points from the mean, indicating that the number of students visiting office hours typically deviates from the mean by about 0.96 students.

Question 5

Which of the following has a higher expected earning?
Option 1: A gift of $240, guaranteed.
Option 2: A 25% chance to win $1,000, and a 75% chance of getting nothing.
A) Option 1
B) Option 2
C) The two expected earnings are equal.

Accepted Answer

The expected earning for Option 2 is calculated as (0.25 * $1,000) + (0.75 * $0) = $250, which is higher than the guaranteed $240 from Option 1.

Question 6

Which of the following has a larger expected loss?
Option 1: A sure loss of $740.
Option 2: A 25% chance to lose nothing, and a 75% chance of losing $1000.
A) Option 1
B) Option 2
C) The two expected earnings are equal.

Accepted Answer

The expected loss for Option 2 is calculated as (0.25 * $0) + (0.75 * $1000) = $750, which is larger than the sure loss of $740 in Option 1.

Question 7

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play).
-Calculate E(X).

Accepted Answer

The answer of Consider a game in which a fair...

Question 8

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play).
-Express Y as a linear transformation of X.
Y = ___(1)____ X + ____(2)_____

Accepted Answer

The answer of Consider a game in which a fair...

Question 9

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play).
-Use the rules for expected value for linear transformations to find E(Y).
$________

Accepted Answer

The answer of Consider a game in which a fair...

Question 10

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play).
-Based on the expected value, is it worth the $5 to enter?
A) No, since the expected profit is also $5.
B) Yes, since in the long-run, your average net profit is greater than zero.

Accepted Answer

In the long run, the player's average net profit is $1.67, which is greater than the $5 entry fee.

Question 11

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Calculate σ_X².

Accepted Answer

The answer of Consider a game in which a fair...

Question 12

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Use the rules for expected value for linear transformations to find σ_Y².

Accepted Answer

The answer of Consider a game in which a fair...

Question 13

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Find σ_Y.

Accepted Answer

The answer of Consider a game in which a fair...

Question 14

Let X be the amount in claims (in dollars) that a randomly chosen policyholder collects from an insurance company this year. From past data, the insurance company has determined that E(X) = $72, and σ_X = $60.
Suppose the insurance company decides to offer a discount to attract new customers. They will pay the new customer $50 for joining, and offer a 5% "cash back" offer for all claims paid. Let Y be the amount in claims (in dollars) for a randomly chosen new customer. Then Y=50+1.05X.
-Find E(Y).

Accepted Answer

The answer of Let X be the amount in claims...

Question 15

Let X be the amount in claims (in dollars) that a randomly chosen policyholder collects from an insurance company this year. From past data, the insurance company has determined that E(X) = $72, and σ_X = $60. Suppose the insurance company decides to offer a discount to attract new customers. They will pay the new customer $50 for joining, and offer a 5% "cash back" offer for all claims paid. Let Y be the amount in claims (in dollars) for a randomly chosen new customer. Then Y=50+1.05X. -Find σ_Y².

Accepted Answer

The answer of Let X be the amount in claims...

Question 16

Let X be the amount in claims (in dollars) that a randomly chosen policyholder collects from an insurance company this year. From past data, the insurance company has determined that E(X) = $72, and σ_X = $60. Suppose the insurance company decides to offer a discount to attract new customers. They will pay the new customer $50 for joining, and offer a 5% "cash back" offer for all claims paid. Let Y be the amount in claims (in dollars) for a randomly chosen new customer. Then Y=50+1.05X. -Find σ_Y.

Accepted Answer

The answer of Let X be the amount in claims...

Question 17

Four friends are contemplating joining a local bowling league. Let XS1U1B11S1U1B0,XS1U1B12S1U1B0,XS1U1B13S1U1B0,XS1U1B14S1U1B0 be the score of the first, second, third, and fourth friend, respectively, on a randomly chosen game. From past experience, the friends know that: E(XS1U1B11S1U1B0 )=110, E(XS1U1B12S1U1B0 )=125, E(XS1U1B13S1U1B0 )=113, and E(XS1U1B14S1U1B0 )=140. Additionally, σS1U1B11S1U1B0=7, σS1U1B11S1U1B0=13, σS1U1B11S1U1B0=10, and σS1U1B11S1U1B0=20. Define their total score on a randomly chosen game as Y=XS1U1B11S1U1B0+XS1U1B12S1U1B0+XS1U1B13S1U1B0+XS1U1B14S1U1B0. Assume the four players' scores are independent.
-The friends decide it would only be worth it to join the bowling league if they could average a total score of 500. Should they join the league?
A) Yes, since their expected total score is greater than 500.
B) No, since their expected total score is less than 500.
C) Yes, since it is possible for their total score to be greater than 500 on a randomly chosen game.
D) No, since it is not possible for their total score to be greater than 500 on a randomly chosen game.

Accepted Answer

The expected total score is the sum of the individual expected scores: 110 + 125 + 113 + 140 = 488, which is less than 500.

Question 18

Four friends are contemplating joining a local bowling league. Let XS1U1B11S1U1B0,XS1U1B12S1U1B0,XS1U1B13S1U1B0,XS1U1B14S1U1B0 be the score of the first, second, third, and fourth friend, respectively, on a randomly chosen game. From past experience, the friends know that: E(XS1U1B11S1U1B0 )=110, E(XS1U1B12S1U1B0 )=125, E(XS1U1B13S1U1B0 )=113, and E(XS1U1B14S1U1B0 )=140. Additionally, σS1U1B11S1U1B0=7, σS1U1B11S1U1B0=13, σS1U1B11S1U1B0=10, and σS1U1B11S1U1B0=20. Define their total score on a randomly chosen game as Y=XS1U1B11S1U1B0+XS1U1B12S1U1B0+XS1U1B13S1U1B0+XS1U1B14S1U1B0. Assume the four players' scores are independent.
-Calculate the variance of their total score on a randomly chosen game.

Accepted Answer

The answer of Four friends are contemplating joining a local...

Question 19

Four friends are contemplating joining a local bowling league. Let XS1U1B11S1U1B0,XS1U1B12S1U1B0,XS1U1B13S1U1B0,XS1U1B14S1U1B0 be the score of the first, second, third, and fourth friend, respectively, on a randomly chosen game. From past experience, the friends know that: E(XS1U1B11S1U1B0 )=110, E(XS1U1B12S1U1B0 )=125, E(XS1U1B13S1U1B0 )=113, and E(XS1U1B14S1U1B0 )=140. Additionally, σS1U1B11S1U1B0=7, σS1U1B11S1U1B0=13, σS1U1B11S1U1B0=10, and σS1U1B11S1U1B0=20. Define their total score on a randomly chosen game as Y=XS1U1B11S1U1B0+XS1U1B12S1U1B0+XS1U1B13S1U1B0+XS1U1B14S1U1B0. Assume the four players' scores are independent. -Calculate σ_Y.

Accepted Answer

The answer of Four friends are contemplating joining a local...

Question 20

Suppose X is a random variable with E(X)=5 and σS1U1B1XS1U1B0S1U1P12S1S1P0=4. Define Y=2X+8.
-Calculate the expected value of Y.

Accepted Answer

The answer of Suppose X is a random variable with...

Quiz 12: Modeling Randomness

Based on her past experience, a professor knows that the probability distribution function for X = number of students who come to her office hours on any given Wednesday is given below. -What is the value of P(X = 4) (the question mark in the table) ?

Based on her past experience, a professor knows that the probability distribution function for X = number of students who come to her office hours on any given Wednesday is given below. -The expected value of X is 1.85. Interpret this value in the context of the problem.

Based on her past experience, a professor knows that the probability distribution function for X = number of students who come to her office hours on any given Wednesday is given below. -The standard deviation of X is 0.96. Interpret this value in the context of the problem.

Which of the following has a higher expected earning? Option 1: A gift of $240, guaranteed. Option 2: A 25% chance to win $1,000, and a 75% chance of getting nothing.

Which of the following has a larger expected loss? Option 1: A sure loss of $740. Option 2: A 25% chance to lose nothing, and a 75% chance of losing $1000.

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Calculate E(X).

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Express Y as a linear transformation of X. Y = _(1) X + (2)___

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play) . -Based on the expected value, is it worth the $5 to enter?

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Calculate σ_X².

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Find σ_Y.

Suppose X is a random variable with E(X)=5 and σ_X²=4. Define Y=2X+8. -Calculate the expected value of Y.

Quiz 12: Modeling Randomness

Based on her past experience, a professor knows that the probability distribution function for X = number of students who come to her office hours on any given Wednesday is given below. -What is the value of P(X = 4) (the question mark in the table) ?

Based on her past experience, a professor knows that the probability distribution function for X = number of students who come to her office hours on any given Wednesday is given below. -The expected value of X is 1.85. Interpret this value in the context of the problem.

Based on her past experience, a professor knows that the probability distribution function for X = number of students who come to her office hours on any given Wednesday is given below. -The standard deviation of X is 0.96. Interpret this value in the context of the problem.

Which of the following has a higher expected earning? Option 1: A gift of $240, guaranteed. Option 2: A 25% chance to win $1,000, and a 75% chance of getting nothing.

Which of the following has a larger expected loss? Option 1: A sure loss of $740. Option 2: A 25% chance to lose nothing, and a 75% chance of losing $1000.

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Calculate E(X).

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play) . -Based on the expected value, is it worth the $5 to enter?

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Calculate σX2.

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Find σY.

Suppose X is a random variable with E(X)=5 and σX2=4. Define Y=2X+8. -Calculate the expected value of Y.

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Calculate σ_X².

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Find σ_Y.

Suppose X is a random variable with E(X)=5 and σ_X²=4. Define Y=2X+8. -Calculate the expected value of Y.