Question 1

A Markov chain has transition matrix   and initial-probability vector   . Find the probability vector for the third stage of the Markov chain. ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)    A

Question 2

If the probability of rain today is 17%, what are the odds against rain today? ​

Accepted Answer

A)  17 : 83 
B)  83 : 17 
C)  20 : 87 
D)  19 : 87 
E)  none of these choices 
A)  17 : 83 
B)  83 : 17 
C)  20 : 87 
D)  19 : 87 
E)  none of these choices B

Question 3

Suppose that a marksman hits the bull's-eye 1 time in 10 shots. If the next 4 shots are independent, find the probability that exactly two of the next 4 shots hit the bull's-eye. ​

Accepted Answer

A)  Pr(two bull's-eyes) =   
B)  Pr(two bull's-eyes) =   
C)  Pr(two bull's-eyes) =   
D)  Pr(two bull's-eyes) =   
E)  Pr(two bull's-eyes) =   
A)  Pr(two bull's-eyes) =   
B)  Pr(two bull's-eyes) =   
C)  Pr(two bull's-eyes) =   
D)  Pr(two bull's-eyes) =   
E)  Pr(two bull's-eyes) =   A

Question 4

Compute   . ​

Accepted Answer

A)  0 
B)  1 
C)  3 
D)  9 
E)  27 
A)  0 
B)  1 
C)  3 
D)  9 
E)  27

Question 5

The probability that daughters of a mother who attends church regularly will also attend church regularly is 0.9, whereas the probability that daughters of a mother who does not attend regularly will attend regularly is 0.2. ​
If a woman attends church regularly and has one daughter, who in turn has one daughter, what is the probability that the granddaughter attends church regularly? Round your answer to two decimal places.
​

Accepted Answer

A)  0.66 
B)  0.42 
C)  0.17 
D)  0.81 
E)  0.83 
A)  0.66 
B)  0.42 
C)  0.17 
D)  0.81 
E)  0.83

Question 6

A ball is drawn from a bag containing 9 red balls numbered 1-9 and 7 white balls numbered 10-16. What is the probability that the ball is white W or odd-numbered O? ​

Accepted Answer

A)  Pr(W∪O) =   
B)  Pr(W∪O) =   
C)  Pr(W∪O) =   
D)  Pr(W∪O) =   
E)  Pr(W∪O) =   
A)  Pr(W∪O) =   
B)  Pr(W∪O) =   
C)  Pr(W∪O) =   
D)  Pr(W∪O) =   
E)  Pr(W∪O) =

Question 7

Suppose that a person groups 18 objects into three groups in such a way that ​
6 objects are in group A.
6 objects are in group B.
7 objects are in group C.
3 objects are in both group A and group B.
5 objects are in both group B and group C.
4 objects are in both group A and group C.
2 objects are in all three groups.
​
What is the probability that an object chosen at random has been placed in group A or group B?
​

Accepted Answer

A)    
B)    
C)    
D)    
E)  0 
A)    
B)    
C)    
D)    
E)  0

Question 8

Compute   . ​

Accepted Answer

A)  6 
B)  13,949 
C)  6,970 
D)  167 
E)  98,770 
A)  6 
B)  13,949 
C)  6,970 
D)  167 
E)  98,770

Question 9

The local community-service funding organization in a certain county has classified the population into those who did not give the previous year, those who gave less than their "fair share," and those who met or exceeded their "fair share." Suppose the organization developed the following transition matrix for these groups. Find the steady-state vector for this county's contribution habits. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 10

A red ball and 19 white balls are in a box. If two balls are drawn, without replacement, what is the probability of getting a red R ball on the first draw and a white W ball on the second? ​

Accepted Answer

A)  Pr(R first ∩ W second) =   
B)  Pr(R first ∩ W second) =   
C)  Pr(R first ∩ W second) =   
D)  Pr(R first ∩ W second) =   
E)  Pr(R first ∩ W second) = 1 
A)  Pr(R first ∩ W second) =   
B)  Pr(R first ∩ W second) =   
C)  Pr(R first ∩ W second) =   
D)  Pr(R first ∩ W second) =   
E)  Pr(R first ∩ W second) = 1

Question 11

Compute   . ​

Accepted Answer

A)  125 
B)  15 
C)  8 
D)  60 
E)  243 
A)  125 
B)  15 
C)  8 
D)  60 
E)  243

Question 12

Find   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 13

A ball is drawn from a bag containing 3 red balls numbered 1-3 and 13 white balls numbered 4-16. What is the probability that the number on the ball is odd O? ​

Accepted Answer

A)  Pr(O) =   
B)  Pr(O) =   
C)  Pr(O) =   
D)  Pr(O) =   
E)  Pr(O) =   
A)  Pr(O) =   
B)  Pr(O) =   
C)  Pr(O) =   
D)  Pr(O) =   
E)  Pr(O) =

Question 14

Five candidates for manager of a department store are ranked according to the weighted average of several criteria. How many different rankings are possible for the top four candidates if no two candidates receive the same rank? ​

Accepted Answer

A)  60 
B)  625 
C)  20 
D)  120 
E)  9 
A)  60 
B)  625 
C)  20 
D)  120 
E)  9

Question 15

An insurance company receives 40 claims in a certain month. Company policy mandates that a random selection of 5 of these claims be thoroughly investigated. If 3 of the 40 claims are fraudulent, find the probability that none of the fraudulent claims are thoroughly investigated. Round your answer to three decimal places. ​

Accepted Answer

A)  0.025 
B)  0.001 
C)  0.662 
D)  0.493 
A)  0.025 
B)  0.001 
C)  0.662 
D)  0.493

Question 16

A ball is drawn from a bag containing 16 red balls numbered 1-16 and 12 white balls numbered 17-28. What is the probability that the ball is white W and odd-numbered O? ​

Accepted Answer

A)  Pr(W∩O) =   
B)  Pr(W∩O) =   
C)  Pr(W∩O) =   
D)  Pr(W∩O) =   
E)  Pr(W∩O) =   
A)  Pr(W∩O) =   
B)  Pr(W∩O) =   
C)  Pr(W∩O) =   
D)  Pr(W∩O) =   
E)  Pr(W∩O) =

Question 17

Suppose an ice cream store offers 36 flavors of ice cream cones in sizes small, medium, and large. How many different selections of cones are possible if each cone has one flavor? ​

Accepted Answer

A)  42,840 
B)  108 
C)  46,656 
D)  7,140 
E)  1,296 
A)  42,840 
B)  108 
C)  46,656 
D)  7,140 
E)  1,296

Question 18

A local business A has two competitors, B and C. No customer patronizes more than one of these businesses at the same time. Initially, the probabilities that a customer patronizes A, B, or C are 0.2, 0.6, and 0.2, respectively. Suppose A initiates an advertising campaign to improve its business and finds the following transition matrix to describe the effect. ​   ​
Find the steady-state vector for this market-that is, the long-range share of the market that each business can expect if the transition matrix holds.
​

Accepted Answer

A)    
B)  ​   
C)    
D)    
E)    
A)    
B)  ​   
C)    
D)    
E)

Question 19

If a pair of dice, one green and one red, is rolled, what is the probability that the sum of the two dice is a number between 5 and 9, inclusive? ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 20

Suppose a die is tossed 1500 times and a 3 comes up 601 times. Find the empirical probability for a 3 to occur. On the basis of a comparison of the empirical probability and the theoretical probability, do you think the die is fair or biased? ​

Accepted Answer

A)  Pr(5) =  
The die is biased since Pr(3) should be about 1/6. 
B)  Pr(5) =  
The die is biased since Pr(3) should be about 1/6. 
C)  Pr(5) =  
The die is biased since Pr(3) should be about 1/6. 
D)  Pr(5) =  
The die is fair since Pr(3) should be about 5/6. 
E)  Pr(5) =  
The die is fair since Pr(3) should be about 5/6. 
A)  Pr(5) =  
The die is biased since Pr(3) should be about 1/6. 
B)  Pr(5) =  
The die is biased since Pr(3) should be about 1/6. 
C)  Pr(5) =  
The die is biased since Pr(3) should be about 1/6. 
D)  Pr(5) =  
The die is fair since Pr(3) should be about 5/6. 
E)  Pr(5) =  
The die is fair since Pr(3) should be about 5/6.

A Markov chain has transition matrix and initial-probability vector . Find the probability vector for the third stage of the Markov chain.

If the probability of rain today is 17%, what are the odds against rain today?

Suppose that a marksman hits the bull's-eye 1 time in 10 shots. If the next 4 shots are independent, find the probability that exactly two of the next 4 shots hit the bull's-eye.

Compute .

A ball is drawn from a bag containing 9 red balls numbered 1-9 and 7 white balls numbered 10-16. What is the probability that the ball is white W or odd-numbered O?

Compute .

A red ball and 19 white balls are in a box. If two balls are drawn, without replacement, what is the probability of getting a red R ball on the first draw and a white W ball on the second?

Compute .

Find .

A ball is drawn from a bag containing 3 red balls numbered 1-3 and 13 white balls numbered 4-16. What is the probability that the number on the ball is odd O?

Five candidates for manager of a department store are ranked according to the weighted average of several criteria. How many different rankings are possible for the top four candidates if no two candidates receive the same rank?

A ball is drawn from a bag containing 16 red balls numbered 1-16 and 12 white balls numbered 17-28. What is the probability that the ball is white W and odd-numbered O?

Suppose an ice cream store offers 36 flavors of ice cream cones in sizes small, medium, and large. How many different selections of cones are possible if each cone has one flavor?

If a pair of dice, one green and one red, is rolled, what is the probability that the sum of the two dice is a number between 5 and 9, inclusive?

Suppose a die is tossed 1500 times and a 3 comes up 601 times. Find the empirical probability for a 3 to occur. On the basis of a comparison of the empirical probability and the theoretical probability, do you think the die is fair or biased?

Linear Equations and Functions

Quadratic and Other Special Functions

Matrices

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Further Topics in Probability and Data Description

Derivatives

Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals - Techniques

A: Definite Integrals - Techniques

Functions of Two or More Variables

Algebraic Concepts 2

Algebraic Concepts 3

Algebraic Concepts 4

Algebraic Concepts 5

Filters

Exam 7: Introduction to Probability

A Markov chain has transition matrix and initial-probability vector . Find the probability vector for the third stage of the Markov chain. ​

If the probability of rain today is 17%, what are the odds against rain today? ​

Suppose that a marksman hits the bull's-eye 1 time in 10 shots. If the next 4 shots are independent, find the probability that exactly two of the next 4 shots hit the bull's-eye. ​

Compute . ​

A ball is drawn from a bag containing 9 red balls numbered 1-9 and 7 white balls numbered 10-16. What is the probability that the ball is white W or odd-numbered O? ​

Compute . ​

A red ball and 19 white balls are in a box. If two balls are drawn, without replacement, what is the probability of getting a red R ball on the first draw and a white W ball on the second? ​

Compute . ​

Find . ​

A ball is drawn from a bag containing 3 red balls numbered 1-3 and 13 white balls numbered 4-16. What is the probability that the number on the ball is odd O? ​

Five candidates for manager of a department store are ranked according to the weighted average of several criteria. How many different rankings are possible for the top four candidates if no two candidates receive the same rank? ​

A ball is drawn from a bag containing 16 red balls numbered 1-16 and 12 white balls numbered 17-28. What is the probability that the ball is white W and odd-numbered O? ​

Suppose an ice cream store offers 36 flavors of ice cream cones in sizes small, medium, and large. How many different selections of cones are possible if each cone has one flavor? ​

If a pair of dice, one green and one red, is rolled, what is the probability that the sum of the two dice is a number between 5 and 9, inclusive? ​

Suppose a die is tossed 1500 times and a 3 comes up 601 times. Find the empirical probability for a 3 to occur. On the basis of a comparison of the empirical probability and the theoretical probability, do you think the die is fair or biased? ​

Linear Equations and Functions

Quadratic and Other Special Functions

Matrices

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Further Topics in Probability and Data Description

Derivatives

Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals - Techniques

A: Definite Integrals - Techniques

Functions of Two or More Variables

Algebraic Concepts 2

Algebraic Concepts 3

Algebraic Concepts 4

Algebraic Concepts 5

Filters

A Markov chain has transition matrix and initial-probability vector . Find the probability vector for the third stage of the Markov chain.

If the probability of rain today is 17%, what are the odds against rain today?

Suppose that a marksman hits the bull's-eye 1 time in 10 shots. If the next 4 shots are independent, find the probability that exactly two of the next 4 shots hit the bull's-eye.

Compute .

A ball is drawn from a bag containing 9 red balls numbered 1-9 and 7 white balls numbered 10-16. What is the probability that the ball is white W or odd-numbered O?

Compute .

A red ball and 19 white balls are in a box. If two balls are drawn, without replacement, what is the probability of getting a red R ball on the first draw and a white W ball on the second?

Compute .

Find .

A ball is drawn from a bag containing 3 red balls numbered 1-3 and 13 white balls numbered 4-16. What is the probability that the number on the ball is odd O?

Five candidates for manager of a department store are ranked according to the weighted average of several criteria. How many different rankings are possible for the top four candidates if no two candidates receive the same rank?

A ball is drawn from a bag containing 16 red balls numbered 1-16 and 12 white balls numbered 17-28. What is the probability that the ball is white W and odd-numbered O?

Suppose an ice cream store offers 36 flavors of ice cream cones in sizes small, medium, and large. How many different selections of cones are possible if each cone has one flavor?

If a pair of dice, one green and one red, is rolled, what is the probability that the sum of the two dice is a number between 5 and 9, inclusive?

Suppose a die is tossed 1500 times and a 3 comes up 601 times. Find the empirical probability for a 3 to occur. On the basis of a comparison of the empirical probability and the theoretical probability, do you think the die is fair or biased?