Deck 12: Modeling Randomness

Question 1

Your brother says he is better than you at playing some video game. You play 6 games and he wins 4 of them. He says that this proves he is a better player. You have studied statistics and you want to determine the probability of anyone winning at least 4 games out of 6 just by chance if you are both equally as skilled. Which of the following would provide an accurate estimate of that probability?

A) Take 6 cards and write "win" on 4 of them and "lose" on 2 of them. Shuffle the cards and place them in two piles and determine if one pile has more "win" cards than the other. Repeat this many times. Calculate the proportion of times one pile had more wins than the other.
B) Toss a six-sided die many times and calculate the proportion of times the number 4 or more lands face up.
C) Flip a coin 6 times and count the number of heads. Repeat this many times. Calculate the proportion of your simulated results that gave 4 or more heads.

Answer 1

Flip a coin 6 times and count the number of heads. Repeat this many times. Calculate the proportion of your simulated results that gave 4 or more heads.

Question 2

There are ten Academic Senate faculty in the Statistics Department at the University of California, Irvine: three females and seven males. Two faculty are to be selected without replacement to serve on a committee where one of those selected will serve as the head of the committee.

-If we record the sex of the chosen committee members (male or female) and their roles (head or not), how many possible outcomes are in the sample space?

A) 1
B) 2
C) 3
D) 4

Answer 2

4

Question 3

There are ten Academic Senate faculty in the Statistics Department at the University of California, Irvine: three females and seven males. Two faculty are to be selected without replacement to serve on a committee where one of those selected will serve as the head of the committee.

-The events in the sample space are all equally likely.

Answer 3

False

Question 4

There are ten Academic Senate faculty in the Statistics Department at the University of California, Irvine: three females and seven males. Two faculty are to be selected without replacement to serve on a committee where one of those selected will serve as the head of the committee.

-How could you simulate this random process?

A) Roll a six-sided die twice. If the first roll is a one, then the committee head is female; if the second roll is a one, then the other committee member is also female.
B) Write male on seven cards and female on three cards. Select one card to be the committee head, and another card to be the other committee member.
C) Flip a coin twice. If the first flip lands on heads, then the committee head is female; if the second coin lands on heads, then the other committee member is also female.

Question 5

There are ten Academic Senate faculty in the Statistics Department at the University of California, Irvine: three females and seven males. Two faculty are to be selected without replacement to serve on a committee where one of those selected will serve as the head of the committee.

-Now, instead of recording the sex and role of the selected committee members, you only record the number of females selected. How many possible outcomes are in this new sample space?

A) 1
B) 2
C) 3
D) 4

Question 6

There are ten Academic Senate faculty in the Statistics Department at the University of California, Irvine: three females and seven males. Two faculty are to be selected without replacement to serve on a committee where one of those selected will serve as the head of the committee.
-A simulation of this random process was run 1,000 times, and the number of females selected for the committee was recorded on each simulation. A bar plot of the results follows.
Use the bar plot to estimate the probability that zero females are selected for the committee.

Question 7

An online psychic (psi) ability or extrasensory perception (ESP) test shows the participant five face-down cards on the screen and asks the participant to "click on the one card that has a picture on the other side." After clicking a card, the test shows you the card to determine if your guess was correct.

-How many possible outcomes are in this sample space?

A) 2
B) 3
C) 4
D) 5

Question 8

An online psychic (psi) ability or extrasensory perception (ESP) test shows the participant five face-down cards on the screen and asks the participant to "click on the one card that has a picture on the other side." After clicking a card, the test shows you the card to determine if your guess was correct.

-If a person does not have ESP, what is the probability the person's guess is correct?

A) 0.10
B) 0.20
C) 0.50
D) 0.80

Question 9

An online psychic (psi) ability or extrasensory perception (ESP) test shows the participant five face-down cards on the screen and asks the participant to "click on the one card that has a picture on the other side." After clicking a card, the test shows you the card to determine if your guess was correct.

-How could you simulate one play of the game, assuming the participant did not have ESP abilities?

A) Flip a coin five times and count the number of heads that appear.
B) Spin a spinner where 1/5^th of the spinner is red and 4/5^ths are white. Record whether you land on the red portion of the spinner or not.
C) Take five cards where one card is black and the other four are white. Shuffle the cards and select one card from the five. Record whether you chose a white card or not.

Question 10

An online psychic (psi) ability or extrasensory perception (ESP) test shows the participant five face-down cards on the screen and asks the participant to "click on the one card that has a picture on the other side." After clicking a card, the test shows you the card to determine if your guess was correct.

-The ESP test website states: "Keep in mind that your results should only be considered suggestive, because high scores in these tests can be due to chance as well as to actual abilities. Only repeated testing can distinguish between luck and genuine psi abilities." Suppose you take the test five times. How would you simulate the probability of choosing the correct card in at least three of the trials, assuming you do not have ESP?

A) Flip a coin five times and count the number of heads that appear. Repeat this many times. Calculate the proportion of your simulated results that gave 3 or more heads.
B) Spin a spinner five times, where 1/5^th of the spinner is red and 4/5^ths are white. Record the number of times the spinner lands on white. Repeat this many times. Calculate the proportion of your simulated results where the spinner landed on white at least three times.
C) Take five cards where one card is black and the other four are white. Shuffle the cards and choose five cards with replacement. Record the number of black cards chosen. Repeat this many times. Calculate the proportion of your simulated results where you chose least three black cards.

Question 11

An online psychic (psi) ability or extrasensory perception (ESP) test shows the participant five face-down cards on the screen and asks the participant to "click on the one card that has a picture on the other side." After clicking a card, the test shows you the card to determine if your guess was correct.

-A simulation of this random process was run 1,000 times, assuming the participant did not have ESP, and the number of correct guesses was recorded on each simulation. A bar plot of the results follows. Using the bar plot, estimate the probability of three or more correct guesses.

A) 0.01
B) 0.04
C) 0.07
D) 0.60

Question 12

An online psychic (psi) ability or extrasensory perception (ESP) test shows the participant five face-down cards on the screen and asks the participant to "click on the one card that has a picture on the other side." After clicking a card, the test shows you the card to determine if your guess was correct.

-Based on the simulation results in the bar plot, if someone guessed correctly four times, would you consider this evidence that the person has ESP?

A) Yes, since the probability the person does not have ESP is less than 1%.
B) Yes, since if they did not have ESP, the probability of four or more correct guesses is less than 1%.
C) No, since they did not guess all five correctly.
D) No, since it's still possible for a person to guess correctly four times if they don't have ESP.

Question 13

According to my neighbor, the probability that the tomato plants she planted last month will actually survive to produce fruit is only 0.6. This means her tomato plants will survive to produce fruit in the next six out of 10 growing seasons.

Question 14

The probability of an event is always equal to the number of outcomes in the event divided by the number of outcomes in the sample space.

Question 15

According to Krantz in his book What the Odds Are (1992, p. 161), the probability that a randomly selected American will be injured by lightning in a given year is 1/685000. What does this probability mean?

A) Every 1 out of 685,000 Americans will be injured by lightning in a given year.
B) If we were to randomly select an American many many times, then the proportion of times that selected American would be injured by lightning in a given year is 1/685000.
C) If we were to randomly select 100 Americans, none of them would be injured by lightning in a given year.
D) Out of 685,000 years, no one will get injured by lightning in 684,999 of those years.

Question 16

Assume that, for events and , and . Are and are mutually exclusive?

A) Yes
B) No
C) We do not have enough information to determine whether and are mutually exclusive.

Question 17

John is taking Statistics and Biology this semester. The probability John will get an A in Statistics is 0.4. The probability John will get an A in Biology is 0.3. The probability John will get an A in both Statistics and Biology is 0.15. What is the probability that John will get at least one A between his Statistics and Biology courses?

A) 0.12
B) 0.15
C) 0.55
D) 0.7

Question 18

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events
R = student lives in a campus residence hall
M = student participates in a campus meal program

-Write the probability 0.42 in probability notation.

A) P(M)
B) P(R)
C) P(R?M)
D) P(R?M)

Question 19

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events
R = student lives in a campus residence hall
M = student participates in a campus meal program

-How would you write the event that a college student "does not participate in a campus meal program" in probability notation?

A) P(M)
B) P(M^C )
C) P(R?M^C )
D) P(R^C ?M)

Question 20

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events
R = student lives in a campus residence hall
M = student participates in a campus meal program

-How would you write the event that a college student "lives on campus and does not participate in a campus meal program" in probability notation?

A) P(M)
B) P(M^C )
C) P(R?M^C )
D) P(R^C ?M)

Question 21

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events
R = student lives in a campus residence hall
M = student participates in a campus meal program
-Find P(R^C).

Question 22

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events
R = student lives in a campus residence hall
M = student participates in a campus meal program
-Find P(M^C).

Question 23

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events
R = student lives in a campus residence hall
M = student participates in a campus meal program
-Use the addition rule to find P(R U M).

Question 24

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events
R = student lives in a campus residence hall
M = student participates in a campus meal program

-Are the events and mutually exclusive?

A) Yes
B) No
C) Not enough information provided

Question 25

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events
R = student lives in a campus residence hall
M = student participates in a campus meal program
-Fill in the probability table below to represent this scenario.

Question 26

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events
R = student lives in a campus residence hall
M = student participates in a campus meal program
-Suppose there are 100 students in this population. Fill in the Venn diagram below to represent the number of students in each category for this scenario. Hint: The four numbers should sum to 100!

Question 27

A lightbulb manufacturing plant's production line has determined that the probability a lightbulb will be defective is 0.04. The quality control team decides to take a random sample of 500 lightbulbs and measure the proportion of those lightbulbs that are defective.
-Use the Normal Probability Calculator applet to approximate the probability the quality control team sees at least 25 defective items in their sample.

Question 28

Suppose you roll a six-sided die once. Define events
A = roll an even number
B = roll a two
C = roll an odd number
D = roll a number greater than three

-Are events B and C mutually exclusive?

Question 29

Suppose you roll a six-sided die once. Define events
A = roll an even number
B = roll a two
C = roll an odd number
D = roll a number greater than three
-Find P(B^C).

Question 30

Suppose you roll a six-sided die once. Define events
A = roll an even number
B = roll a two
C = roll an odd number
D = roll a number greater than three
-Find P(A∪B).

Question 31

Suppose you roll a six-sided die once. Define events
A = roll an even number
B = roll a two
C = roll an odd number
D = roll a number greater than three
-Find P(C∪D).

Question 32

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.39, and winning is independent from one bottle to the next. You buy six bottles. What is the probability that none of the six bottles reveals a prize?

A) 0.39
B) 0.61
C) 0.052
D) 0.004

Question 33

Assume that events A and B are independent, where P(A)=0.3 and P(B)=0.6. What is P(A|B)?

A) 0.6
B) 0.3
C) 0.18
D) 0

Question 34

Assume that events and are mutually exclusive, where and . What is ?

A) 0.6
B) 0.3
C) 0.18
D) 0

Question 35

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events
R = student lives in a campus residence hall
M = student participates in a campus meal program

-Are the events and independent?

A) Yes
B) No
C) Not enough information provided

Question 36

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events
R = student lives in a campus residence hall
M = student participates in a campus meal program

-What is the appropriate expression for the probability that a randomly chosen college student who lives in a campus residence hall participates in a campus meal program?

A) P(R?M)
B) P(M?R)
C) P(M?R)
D) P(M?R)

Question 37

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events
R = student lives in a campus residence hall
M = student participates in a campus meal program
-Find P(R│M)..

Question 38

Suppose that 70% of the seniors at a large university have taken calculus, and 30% of the seniors have taken physics. Of the seniors who have taken calculus, 40% have taken physics. A student who is a senior at this university is randomly selected. Define the following events.
A = The student has taken calculus
B = The student has taken physics

-Which of the following is true about the events A and B?

A) A and B are mutually exclusive.
B) A and B are complements.
C) A and B are independent.
D) None of the above.

Question 39

Suppose that 70% of the seniors at a large university have taken calculus, and 30% of the seniors have taken physics. Of the seniors who have taken calculus, 40% have taken physics. A student who is a senior at this university is randomly selected. Define the following events.
A = The student has taken calculus
B = The student has taken physics

-Find P(A?B)

A) 0.70 + 0.30 - 0.40 = 0.6
B) (0.70)(0.30) = 0.21
C) (0.70)(0.40) = 0.28
D) (0.30)(0.40) = 0.12

Question 40

Data on sex and opinion on the death penalty for respondents in the 2008 General Social Survey in which a random sample of 1,902 adults in the U.S. was surveyed are below.

-What is the probability that the respondent will oppose the death penalty, given that the respondent is male?

A) 254/1902
B) 639/1902
C) 254/639
D) 254/885

Question 41

Data on sex and opinion on the death penalty for respondents in the 2008 General Social Survey in which a random sample of 1,902 adults in the U.S. was surveyed are below.

-What is the probability that the respondent is female and opposes the death penalty?

A) 385/1902
B) 385/639
C) 385/1017
D) 639/1902

Question 42

Suppose that only 8% of a large population has a certain disease. A diagnostic test has been developed which is 90% accurate for people with the disease (90% of people with the disease test positive), and 85% accurate for people without the disease (85% of people without the disease test negative). Define the following events:
A = person has the disease
B = person tests positive on the diagnostic test
-For each of the given probabilities, state whether the probability is a conditional probability or an unconditional probability:
0.08?
0.90?
0.85?

Question 43

Suppose that only 8% of a large population has a certain disease. A diagnostic test has been developed which is 90% accurate for people with the disease (90% of people with the disease test positive), and 85% accurate for people without the disease (85% of people without the disease test negative). Define the following events:
A = person has the disease
B = person tests positive on the diagnostic test

-How would you express the probability 0.90 in terms of A and B?

A) P(A)
B) P(B)
C) P(A?B)
D) P(B?A)
E) P(A?B) F. P(A?B)

Question 44

Suppose that only 8% of a large population has a certain disease. A diagnostic test has been developed which is 90% accurate for people with the disease (90% of people with the disease test positive), and 85% accurate for people without the disease (85% of people without the disease test negative). Define the following events:
A = person has the disease
B = person tests positive on the diagnostic test
-Fill in the values below to create a hypothetical table of 1,000 randomly selected individuals in this population.

Question 45

Suppose that only 8% of a large population has a certain disease. A diagnostic test has been developed which is 90% accurate for people with the disease (90% of people with the disease test positive), and 85% accurate for people without the disease (85% of people without the disease test negative). Define the following events:
A = person has the disease
B = person tests positive on the diagnostic test
-Fill in the tree diagram below to represent this scenario.

Question 46

Suppose that only 8% of a large population has a certain disease. A diagnostic test has been developed which is 90% accurate for people with the disease (90% of people with the disease test positive), and 85% accurate for people without the disease (85% of people without the disease test negative). Define the following events:
A = person has the disease
B = person tests positive on the diagnostic test
-What is the probability that a randomly selected person tests positive on the diagnostic test?

Question 47

Suppose that only 8% of a large population has a certain disease. A diagnostic test has been developed which is 90% accurate for people with the disease (90% of people with the disease test positive), and 85% accurate for people without the disease (85% of people without the disease test negative). Define the following events:
A = person has the disease
B = person tests positive on the diagnostic test
-Given that a person tests positive on the diagnostic test, what is the probability he or she has the disease?

Question 48

Which of the following variables is a discrete random variable?

A) Time it takes one email to travel between a sender and receiver
B) Number of letters in the last name of a randomly chosen student
C) Weight a dieter will lose after following a two-week weight loss program
D) High temperature in Irvine on a randomly chosen day

Question 49

Which of the following variables is a discrete random variable?

A) Time it takes a randomly selected student to finish this exam
B) Body temperature of a randomly selected adult
C) Distance a randomly selected adult walks in a day
D) Number of texts a randomly selected college student receives in a day

Question 50

The standard deviation of a random variable Y is the…

A) range of possible values for Y.
B) most common value over a large number of observations of Y.
C) approximate average distance from the mean that one would see in a large number of observations of Y.
D) approximate mean value over a large number of observations of Y.

Question 51

Let X be the amount in claims (in dollars) that a randomly chosen policyholder collects from an insurance company this year. The probability distribution function of X is given below.

-What is the probability that a randomly chosen policy holder claims more than $100?

A) 0.07
B) 0.13
C) 0.20
D) 0.87

Question 52

Let X be the amount in claims (in dollars) that a randomly chosen policyholder collects from an insurance company this year. The probability distribution function of X is given below.
-Calculate the expected value of X.
$_________

Question 53

Let X be the amount in claims (in dollars) that a randomly chosen policyholder collects from an insurance company this year. The probability distribution function of X is given below.

-Let $D be the expected value of X. Which of the following is a correct interpretation of this value?

A) We would expect a randomly chosen policyholder to collect $D in claims this year.
B) The most likely value for the amount in claims is $D.
C) If we were to observe a large number of policyholders, the average amount claimed would be around $D.
D) On average, the amount claimed is about $D away from the mean.

Question 54

Let X be the amount in claims (in dollars) that a randomly chosen policyholder collects from an insurance company this year. The probability distribution function of X is given below.
-Calculate the standard deviation of X.
$________

Question 55

Let X be the amount in claims (in dollars) that a randomly chosen policyholder collects from an insurance company this year. The probability distribution function of X is given below.

-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 policyholder to collect $S in claims this year.
B) The most likely value for the amount in claims is $S.
C) If we were to observe a large number of policyholders, the average amount claimed would be around $S.
D) On average, the amount claimed is about $S away from the mean.

Question 56

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.)
-Determine the probability distribution of X:

Question 57

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.)
-Calculate E(X).
$_______

Question 58

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 $D be the expected value 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.

Question 59

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.)
-Fill in the blank with the correct dollar amount:
If the lottery agency does not want to lose money over the long run, they must charge at least
$_______ per ticket.

Question 60

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.)
-Calculate the standard deviation of X.
$________

Question 61

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.

Question 62

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

Question 63

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.

Question 64

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.

Question 65

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.

Question 66

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.

Question 67

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).

Question 68

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)_____

Question 69

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).
$________

Question 70

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.

Question 71

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.

Question 72

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.

Question 73

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.

Question 74

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).

Question 75

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.

Question 76

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.

Question 77

Four friends are contemplating joining a local bowling league. Let X₁,X₂,X₃,X₄ 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(X₁ )=110, E(X₂ )=125, E(X₃ )=113, and E(X₄ )=140. Additionally, ?₁=7, ?₁=13, ?₁=10, and ?₁=20. Define their total score on a randomly chosen game as Y=X₁+X₂+X₃+X₄. 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.

Question 78

Four friends are contemplating joining a local bowling league. Let X₁,X₂,X₃,X₄ 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(X₁ )=110, E(X₂ )=125, E(X₃ )=113, and E(X₄ )=140. Additionally, σ₁=7, σ₁=13, σ₁=10, and σ₁=20. Define their total score on a randomly chosen game as Y=X₁+X₂+X₃+X₄. Assume the four players' scores are independent.
-Calculate the variance of their total score on a randomly chosen game.

Question 79

Four friends are contemplating joining a local bowling league. Let X₁,X₂,X₃,X₄ 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(X₁ )=110, E(X₂ )=125, E(X₃ )=113, and E(X₄ )=140. Additionally, σ₁=7, σ₁=13, σ₁=10, and σ₁=20. Define their total score on a randomly chosen game as Y=X₁+X₂+X₃+X₄. Assume the four players' scores are independent.
-Calculate σ_Y.

Question 80

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