Question 1

The random variable X is known to be uniformly distributed between 2 and 12. Compute E(X), the expected value of the distribution.
A)4
B)5
C)6
D)7

Accepted Answer

D

Question 2

The _______________ probability distribution can be used to estimate the number of vehicles that go through an intersection during the lunch hour. ​​
A)binomial
B)normal
C)triangular
D)Poisson

Accepted Answer

The Poisson distribution is commonly used to model the number of events that occur in a fixed interval of time or space, where the events occur randomly and independently at a constant rate. This makes it suitable for estimating the number of vehicles that pass through an intersection during a fixed time interval, such as the lunch hour. The other distributions listed (binomial, normal, and triangular) are not appropriate for modeling this type of situation.

Question 3

The event containing the outcomes belonging to A or B or both is the ________________ of A and B.
A)union
B)Venn diagram
C)intersection
D)subset

Accepted Answer

The union of A and B represents the event containing all outcomes belonging to A or B or both. This is represented by the symbol ∪.

Question 4

A(n) ________________ is a description of how probabilities are distributed over the values of a random variable.​
A)​probability distribution
B)​mass function of an event
C)​density function
D)​expected value

Accepted Answer

A probability distribution is specifically defined as a description of how probabilities are distributed over the values of a random variable. The other options listed, such as the mass function of an event, density function, and expected value, are all related concepts but do not fully capture the definition of a probability distribution.

Question 5

A joint probability is the
A)sum of the probabilities of two events.
B)probability of the intersection of two events.
C)probability of the union of two events.
D)sum of the probabilities of two independent events.

Accepted Answer

Joint probability refers to the probability of both events occurring simultaneously. It is calculated as the intersection of the two events, which is the common outcome or overlap between them. Therefore, option B is the correct choice. Option A is incorrect as it refers to the sum of probabilities of two events which can be mutually exclusive or inclusive. Option C is incorrect as it refers to the probability of either event occurring, not both. Option D is incorrect as it assumes independence between the events, which may not always be the case.

Question 6

The random variable X is known to be uniformly distributed between 2 and 12. Compute the standard deviation of X.
A)​2.887
B)​3.464
C)8.333
D)​12

Accepted Answer

The answer of The random variable X is known to...

Question 7

Sample space is
A)​a process that results in some outcome.
B)​the collection of all possible outcomes.
C)the collection of events
D)​a subgroup of a population/the likelihood of an outcome.

Accepted Answer

The sample space is the collection of all possible outcomes that can result from a particular process or experiment.

Question 8

In the probability table below, which value is a marginal probability? ?? $$\begin{array} { | l | l | l | c | } 
\hline &  { \text { Completed } } \
\hline \text { Obstacle Course Level } & \text { No } & \text { Yes } & \text { Total } \
\hline \text { Challenging } & 0.4 & 0.3 & 0.7 \
\hline \text { Easy } & 0.1 & 0.2 & 0.3 \
\hline \text { Total } & 0.5 & 0.5 & 1 \
\hline
\end{array}$$
A)0.1
B)1
C)0.5
D)0.4

Accepted Answer

The marginal probability is the probability of an event occurring, regardless of any other events. In this case, the marginal probability of completing the obstacle course is 0.5, which is the sum of the probabilities of completing the course at each level.

Question 9

Two events are independent if
A)the two events occur at the same time.
B)the probability of one or both events is greater than 1.
C)the probability of each event is not affected by the other.
D)none of these.

Accepted Answer

Independence means that the occurrence of one event does not affect the probability of the other event. Therefore, the correct option is C, which states that the probability of each event is not affected by the other. Options A and B are incorrect as they do not relate to the concept of independence, and option D is irrelevant.

Question 10

An initial estimate of the probabilities of events is a ______________ probability.
A)​posterior
B)​conditional
C)empirical
D)​prior

Accepted Answer

An initial estimate of the probabilities of events is a prior probability.

Question 11

A variable that can only take on specific numeric values is called a ​​
A)categorical variable.
B)discrete random variable.
C)continuous random variable.

Accepted Answer

A variable that can only take on specific numeric values is an example of a discrete random variable, as it can only take on a finite or countable number of possible values. Categorical variables refer to variables that consist of categories or groups, and continuous random variables can take on any value within a specified range.

Question 12

All the events in the sample space that are not part of the specified event are called
A)joint events.
B)the complement of the event.
C)simple events.
D)independent events.

Accepted Answer

The complement of an event consists of all the outcomes in the sample space that are NOT part of the specified event. Therefore, option B is the correct choice. Joint events refer to two or more events that occur simultaneously, but this is not relevant to the question. Simple events refer to individual outcomes in the sample space. Independent events refer to events where the occurrence of one event does not affect the probability of another event occurring, but this is also not applicable to the question.

Question 13

If a​ z-score is​ zero, then the corresponding x-value must be equal to the ​​
A)mean.
B)median.
C)mode.
D)standard deviation.

Accepted Answer

When a z-score is equal to zero, it means that the corresponding x-value is equal to the mean of the data set. This is because the z-score formula subtracts the mean from the x-value and divides by the standard deviation. If the z-score is zero, then the x-value minus the mean is zero, meaning that the x-value must be equal to the mean.

Question 14

Probability is​ the
A)number of successes divided by the number of failures.
B)numerical measure likelihood that an outcome occurs.
C)chance that an event will not happen.
D)number of successes divided by the standard deviation of the distribution.

Accepted Answer

Probability is a numerical measure of the likelihood that an event will occur. It ranges from 0 (impossible) to 1 (certain). A probability of 0.5 indicates that the event is equally likely to occur or not occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Question 15

All of the following are examples of discrete random variables except ​​
A)number of tickets sold.
B)marital status.
C)time.
D)population of a city.

Accepted Answer

Time is a continuous random variable because it can take on any value within an interval and is not countable.

Question 16

Which of the following is a discrete random variable?
A)The number of times a student guesses the answers to questions on a certain test.
B)The amount of gasoline purchased by a customer.
C)​The amount of mercury found in fish caught in the Gulf of Mexico.
D)​The height of water-oak trees.

Accepted Answer

A discrete random variable is one that has a countable number of possible outcomes, where each outcome has a positive probability of occurrence. In this case, the number of times a student guesses answers on a test can only be a whole number and each guess is a separate outcome that can be counted. Therefore, it is a discrete random variable. The other options are continuous random variables, as they can take on any value within a certain range (e.g. any amount of gasoline, any amount of mercury in fish, any height of a water-oak tree).

Question 17

Which statement is true about mutually exclusive events?
A)If events A and B cannot occur at the same time, they are called mutually exclusive.
B)If either event A or event B must occur, they are called mutually exclusive.
C)P(A) + P(B) = 1 for any events A and B that are mutually exclusive.
D)None of these.

Accepted Answer

Mutually exclusive events are events that cannot occur at the same time or they have no outcomes in common. Therefore, statement A is true. Statement B is incorrect because mutually exclusive events cannot both occur. Statement C is incorrect because the sum of probabilities for mutually exclusive events is not necessarily equal to one. Therefore, the correct option is A.

Question 18

Which of the following statements is correct?
A)The binomial and normal distributions are both discrete probability distributions.
B)The binomial and normal distributions are both continuous probability distributions.
C)The binomial distribution is a continuous probability distribution and the normal distribution is a discrete probability distribution.
D)The binomial distribution is a discrete probability distribution and the normal distribution is a continuous probability distribution.

Accepted Answer

The binomial distribution is a discrete probability distribution because it describes the number of successes in a fixed number of trials. The normal distribution is a continuous probability distribution because it describes a continuous variable, such as height or weight, where the values can take on any value within a certain range.

Question 19

Bayes' theorem is a method used to compute ___________________ probabilities.
A)​posterior
B)​conditional
C)empirical
D)​prior

Accepted Answer

Bayes' theorem is used to compute posterior probabilities which are the revised probabilities after new evidence or information is taken into account. It is also used to calculate conditional probabilities which provide the likelihood of an event occurring given that another event has already occurred.

Question 20

An experiment consists of determining the speed of automobiles on a highway by the use of radar equipment. The random variable in this experiment is a
A)discrete random variable.
B)continuous random variable.
C)complex random variable.
D)categorical random variable.

Accepted Answer

Since the speed of automobiles on the highway can take on any value within a certain range (e.g. 0-100 mph), it is a continuous random variable.

Quiz 5: Probability: an Introduction to Modeling Uncertainty

The random variable X is known to be uniformly distributed between 2 and 12. Compute E(X) , the expected value of the distribution.

The _______________ probability distribution can be used to estimate the number of vehicles that go through an intersection during the lunch hour.

The event containing the outcomes belonging to A or B or both is the ________________ of A and B.

A(n) ________________ is a description of how probabilities are distributed over the values of a random variable.

A joint probability is the

The random variable X is known to be uniformly distributed between 2 and 12. Compute the standard deviation of X.

Sample space is

Two events are independent if

An initial estimate of the probabilities of events is a ______________ probability.

A variable that can only take on specific numeric values is called a

All the events in the sample space that are not part of the specified event are called

If a z-score is zero, then the corresponding x-value must be equal to the

Probability is the

All of the following are examples of discrete random variables except

Which of the following is a discrete random variable?

Which statement is true about mutually exclusive events?

Which of the following statements is correct?

Bayes' theorem is a method used to compute ___________________ probabilities.

An experiment consists of determining the speed of automobiles on a highway by the use of radar equipment. The random variable in this experiment is a