Question 1

A false negative in screening tests is the event that the test is positive for a given condition, given that the person does not have the condition.

Accepted Answer

A false negative in screening tests is the event that the test is negative for a given condition, given that the person actually has the condition.

Question 2

A false positive in screening (e.g., home pregnancy tests) represents the event that:
A) the test is negative for a given condition, given that the person does not have the condition
B) the test is positive for a given condition, given that the person does not have the condition
C) the test is negative for a given condition, given that the person has the condition
D) the test is positive for a given condition, given that the person has the condition

Accepted Answer

A false positive occurs when the test indicates the presence of a condition (in this case pregnancy), when in fact the person does not have the condition. Therefore, the correct option is B, "the test is positive for a given condition, given that the person does not have the condition".

Question 3

Let x be a random variable with the following distribution:
  Round your answer to three decimal places, if necessary. 
a. Find the expected value of x.
______________
b. Find the standard deviation of x.
______________
c. What is the probability that x is further than one standard deviation from the mean?
______________

Accepted Answer

The answer of Let x be a random variable with...

Question 4

A false positive in screening tests is the event that the test is negative for a given condition, given that the person has the condition.

Accepted Answer

A false positive in screening tests is the event that the test is positive for a given condition, even though the person does not actually have the condition.

Question 5

From experience, a shipping company knows that the cost of delivering a small package within 24 hours is $16.20. The company charges $16.95 for shipment but guarantees to refund the charge if delivery is not made within 24 hours. If the company fails to deliver only 3% of its packages within the 24-hour period, what is the expected gain per package?
______________

Accepted Answer

The answer of From experience, a shipping company knows that...

Question 6

A false negative in screening tests (e.g., Steroid testing of athletes) represents the event that:
A) the test is negative for a given condition, given that the person does not have the condition
B) the test is positive for a given condition, given that the person does not have the condition
C) the test is negative for a given condition, given that the person has the condition
D) the test is positive for a given condition, given that the person has the condition

Accepted Answer

A false negative means that a person has the condition, but the test results are negative. Therefore, option C is the correct choice. Option A describes a true negative, option B describes a false positive, and option D describes a true positive.

Question 7

Which of the following statements is false?
A) The mean of a discrete probability distribution is equal to the square root of the variance.
B) The expected value of a discrete probability distribution is the long-run average value if the experiment is to be repeated many times.
C) The standard deviation of a discrete probability distribution measures the average variation of the random variable from the mean.
D) All of the above.
E) None of these.

Accepted Answer

The statement that the mean of a discrete probability distribution is equal to the square root of the variance is false. The mean of a discrete probability distribution is equal to the expected value, while the variance is the expected value of the squared deviations from the mean. The standard deviation is equal to the square root of the variance.

Question 8

Approximately 5% of the bolts coming off a production line have serious defects. Two bolts are randomly selected for inspection. Find the probability distribution for x, the number of defective bolts in the sample.
a. P(x = 0) = P(both bolts are not defective)
______________
b. P(x = 1) = P(one bolt is defective)
______________
c. P(x = 2) = P(both bolts are defective)
______________
d. Find E(x).
______________
e. Find
  .
______________

Accepted Answer

The answer of Approximately 5% of the bolts coming off...

Question 9

Steve takes either a bus or the subway to go to work with probabilities 0.25 and 0.75, respectively. When he takes the bus, he is late 40% of the days. When he takes the subway, he is late 30% of the days. If Steve is late for work on a particular day, what is the probability that he took the bus?
______________

Accepted Answer

The answer of Steve takes either a bus or the...

Question 10

The random variable x is defined as the number of mistakes made by a typist on a randomly chosen page of a physics thesis. The probability distribution follows.
  Round your answer to four decimal places, if necessary. 
a. What is E(x)?
______________
b. Find
  .
______________ 
c. Find P(x < 1).
______________
d. Find
  .
______________ 
e. In what fraction of pages in the thesis would the number of mistakes made be within two standard deviations of the mean?
______________

Accepted Answer

The answer of The random variable x is defined as...

Question 11

Medical case histories indicate that different illnesses may produce identical symptom. Suppose a particular set of symptoms, which we will denote as event H, occurs only when any one of these illnesses: A, B, or C occurs. (For the sake of simplicity, we will assume that illnesses A, B, and C are mutually exclusive.) Studies show these probabilities of getting the three illnesses:  The probabilities of developing the symptoms H, given a specific illness, are:  Assuming that an ill person shows the symptoms H, what is the probability that the person has illness A?
Round your answer to four decimal places.
______________

Accepted Answer

The answer of Medical case histories indicate that different illnesses...

Question 12

Bayes' Rule is a formula for revising an initial subjective (prior) probability value on the basis of results obtained by an empirical investigation and for, thus, obtaining a new (posterior) probability value.

Accepted Answer

This is an accurate definition of Bayes' Rule.

Question 13

In the "Quick Draw" casino card game, a player chooses one card from a deck of 52 well-shuffled cards. If the card the player selects is the king of hearts, the player wins $104; if the card is an ace, the player wins $78; if the card selected is anything else, the player loses $13. Considering a negative amount won to be a loss, how much should the player expect to win in one play of "Quick Draw"?
$ ______________

Accepted Answer

The answer of In the "Quick Draw" casino card game,...

Question 14

Screening tests (e.g., AIDS testing) are evaluated on the probability of a false negative or a false positive, and both of these are:
A) conditional probabilities
B) probability of the intersection of two events
C) probability of the union of two events
D) marginal probabilities
E) probability of dependent events

Accepted Answer

False negative and false positive results are both conditional probabilities, meaning they are the probability of one event (the test result) given that another event (the true status of the individual being tested) has occurred.

Question 15

The neighborhood deli specializes in New York Style Cheesecake. The cheesecakes are made fresh daily and any unsold cheesecake is donated to the Salvation Army soup kitchen to feed the homeless. Each cheesecake costs $5 to make and sells for $11. The daily demand for cheesecake (the number the deli could sell if they had a cheesecake for everyone who wanted one) has the following distribution.
  
a. What is the expected demand for cheesecake?
______________
b. What is the expected daily profit if the deli makes only one cheesecake per day?
$ ______________
c. What is the expected daily profit if the deli makes only two cheesecakes per day?
$ ______________

Accepted Answer

The answer of The neighborhood deli specializes in New York...

Question 16

A sample is selected from one of two populations, S₁ and S₂, with probabilities P(S₁) = 0.80 and P(S₂) = 0.20. If the sample has been selected from S₁, the probability of observing an event A is P(A / S₁) = 0.15. Similarly, If the sample has been selected from S₂, the probability of observing A is P(A / S₂) = 0.25. Round your answer to four decimal places, if necessary a. If a sample is randomly selected from one of the two populations, what is the probability that event A occurs? ______________ If a sample is randomly selected and event A is observed, what is the probability that the sample was selected: b. From population S₁? ______________ c. From population S₂? ______________

Accepted Answer

The answer of A sample is selected from one of...

Question 17

Let the random variable x represent the number of cars owned by a family. Assume that x can take on five values: 0, 1, 2, 3, 4. A partial probability distribution is shown below:
  
a. Find the probability that a family owns three cars.
______________
b. Find the probability that a family owns between 1 and 3 cars, inclusive.
______________
c. Find the probability that a family owns no more than one car.
______________
d. What is the probability that a family owns more than two cars?
______________
e. What is the probability that a family owns at most 3 cars?
______________

Accepted Answer

The answer of Let the random variable x represent the...

Question 18

The probability distribution of your winnings at a casino's card game is shown below.
  
a. What is the chance you win more than $1 if you play just once?
______________
b. How much should you expect to win if you play the game once?
______________
c. After breaking the bank at the casino playing this card game, you decide to open your own casino where the customers can play your favorite card game. How much should you charge the players if you want to have an average profit of $1 per play?
______________

Accepted Answer

The answer of The probability distribution of your winnings at...

Question 19

Let x denote the weight gain in pounds per month for a calf. The probability distribution of x is shown below.
  
a. Find the average weight gain in pounds per month for a calf.
______________
b. Find the variance of the weight gain.
______________
c. What is
  ?
______________ 
d. What is
  ?
______________ 
e. What is
  ?
______________

Accepted Answer

The answer of Let x denote the weight gain in...

Question 20

Given a set of events  that are mutually exclusive and exhaustive and an event A, the law of total probability states that P(A) can be expressed as  .

Accepted Answer

The law of total probability states that if we have a set of events that are mutually exclusive and exhaustive, then the probability of any event A can be expressed as the sum of the probabilities of A given each of the events in the set, weighted by the probability of each event in the set. Therefore, the statement is true.

Quiz 4: Probability and Probability Distributions

A false negative in screening tests is the event that the test is positive for a given condition, given that the person does not have the condition.

A false positive in screening (e.g., home pregnancy tests) represents the event that:

Let x be a random variable with the following distribution: Round your answer to three decimal places, if necessary. a. Find the expected value of x. b. Find the standard deviation of x. c. What is the probability that x is further than one standard deviation from the mean? __

A false positive in screening tests is the event that the test is negative for a given condition, given that the person has the condition.

A false negative in screening tests (e.g., Steroid testing of athletes) represents the event that:

Which of the following statements is false?

Bayes' Rule is a formula for revising an initial subjective (prior) probability value on the basis of results obtained by an empirical investigation and for, thus, obtaining a new (posterior) probability value.

Screening tests (e.g., AIDS testing) are evaluated on the probability of a false negative or a false positive, and both of these are:

Let x denote the weight gain in pounds per month for a calf. The probability distribution of x is shown below. a. Find the average weight gain in pounds per month for a calf. b. Find the variance of the weight gain. c. What is ? d. What is ? e. What is ? __

Given a set of events that are mutually exclusive and exhaustive and an event A, the law of total probability states that P(A) can be expressed as .