Question 1

Compute   . ​

Accepted Answer

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

Question 2

A mathematics class consists of 17 engineering majors E, 13 science majors S, and 9 liberal arts majors LA. Twelve of the engineering students, 3 of the science majors, and 3 of the liberal arts majors are female. What is the probability that a student selected at random is an engineering major or is a female F? ​

Accepted Answer

A) Pr(E∪F) =   
B) Pr(E∪F) =   
C) Pr(E∪F) =   
D) Pr(E∪F) =   
E) Pr(E∪F) =   
A) Pr(E∪F) =   
B) Pr(E∪F) =   
C) Pr(E∪F) =   
D) Pr(E∪F) =   
E) Pr(E∪F) =

Question 3

Suppose that a marksman hits the bull's-eye 1 time in 40 shots. If the next 4 shots are independent, find the probability that all of the next 4 shots hit the bull's-eye. ​

Accepted Answer

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

Question 4

In how many ways can a hand consisting of 4 spades, 3 hearts, 2 clubs, and 1 diamond be selected from a deck of 52 cards? ​

Accepted Answer

A) 86,248,800 
B) 464,199,736 
C) 207,352,860 
D) 3,421,322,190 
E) 9,425,130 
A) 86,248,800 
B) 464,199,736 
C) 207,352,860 
D) 3,421,322,190 
E) 9,425,130

Question 5

A bag contains 5 white balls and 5 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn,with replacement, one ball is red R and one is white W. ​

Accepted Answer

A) Pr(one R ∩one W) =   
B) Pr(one R ∩one W) =   
C) Pr(one R ∩one W) =   
D) Pr(one R ∩one W) =   
E) Pr(one R ∩one W) =   
A) Pr(one R ∩one W) =   
B) Pr(one R ∩one W) =   
C) Pr(one R ∩one W) =   
D) Pr(one R ∩one W) =   
E) Pr(one R ∩one W) =

Question 6

Forty-eight percent of the U.S. population is female, and 23.3% of the female population have a college degree. If a U.S. resident is chosen at random, what is the probability, to three decimal places, that the person is a female with a college degree? ​

Accepted Answer

A) Pr(female ∩ college degree) = 0.233 
B) Pr(female ∩ college degree) = 0.713 
C) Pr(female ∩ college degree) = 0.023 
D) Pr(female ∩ college degree) = 0.112 
E) Pr(female ∩ college degree) = 0.048 
A) Pr(female ∩ college degree) = 0.233 
B) Pr(female ∩ college degree) = 0.713 
C) Pr(female ∩ college degree) = 0.023 
D) Pr(female ∩ college degree) = 0.112 
E) Pr(female ∩ college degree) = 0.048

Question 7

How many four-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if the numbers are less than 9,000 and digits are not used repeatedly? (Hint: Begin with the digit where there is a restriction on the choices.) ​

Accepted Answer

A) 43 
B) 626 
C) 15,625 
D) 300 
E) 15,655 
A) 43 
B) 626 
C) 15,625 
D) 300 
E) 15,655

Question 8

A ball is drawn from a bag containing 15 red balls numbered 1-15 and 6 white balls numbered 16-21. What is the probability that the ball is red R and even-numbered E? ​

Accepted Answer

A) Pr(R∩
E)  =   
B) Pr(R∩
E)  =   
C) Pr(R∩
E)  =   
D) Pr(R∩
E)  =   
E) Pr(R∩
E)  =   
A) Pr(R∩
E)  =   
B) Pr(R∩
E)  =   
C) Pr(R∩
E)  =   
D) Pr(R∩
E)  =   
E) Pr(R∩
E)  =

Question 9

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 10

A company selling substandard drugs to developing countries sold 2,500,000 capsules with 150,000 of them empty. What is the probability that a person who takes a randomly chosen capsule from this company will get an empty capsule? ​

Accepted Answer

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

Question 11

An urn contains three red R balls numbered 1, 2 and 3, four white W balls numbered 4, 5, 6 and 7, and three black B balls numbered 8, 9 and 10 A ball is drawn from the urn. What is the probability that it is red and odd-numbered O? ​

Accepted Answer

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

Question 12

A bag contains 3 nickels, 5 dimes, and 4 quarters. If you draw 3 coins at random from the bag, without replacement, what is the probability that you will get a nickel N, a quarter Q, and a nickel N, in that order? ​

Accepted Answer

A) Pr(N first ∩ Q second ∩ N third) =   
B) Pr(N first ∩ Q second ∩ N third) =   
C) Pr(N first ∩ Q second ∩ N third) =   
D) Pr(N first ∩ Q second ∩ N third) =   
E) Pr(N first ∩ Q second ∩ N third) =   
A) Pr(N first ∩ Q second ∩ N third) =   
B) Pr(N first ∩ Q second ∩ N third) =   
C) Pr(N first ∩ Q second ∩ N third) =   
D) Pr(N first ∩ Q second ∩ N third) =   
E) Pr(N first ∩ Q second ∩ N third) =

Question 13

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 14

A box contains 3 red balls, 5 white balls, and 3 black balls. Two balls are drawn at random from the box with replacement of the first before the second is drawn.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) =   
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) =

Question 15

A red ball and 17 white balls are in a box. If two balls are drawn, without replacement, what is the probability of getting 2 red R balls? ​

Accepted Answer

A) Pr(R first ∩ R second) = 1 
B) Pr(R first ∩ R second) =   
C) Pr(R first ∩ R second) =   
D) Pr(R first ∩ R second) =   
E) Pr(R first ∩ R second) = 0 
A) Pr(R first ∩ R second) = 1 
B) Pr(R first ∩ R second) =   
C) Pr(R first ∩ R second) =   
D) Pr(R first ∩ R second) =   
E) Pr(R first ∩ R second) = 0

Question 16

Two balls are drawn, without replacement, from a bag containing 18 blue balls numbered 1-18 and 4 green balls numbered 19-22. What is the probability that the first ball is blue and even-numbered and the second ball is even-numbered? ​

Accepted Answer

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

Question 17

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)

Question 18

An unprepared student must take a 7-question, multiple-choice test that has 5 possible answers per question. If the student can eliminate 2 of the possible answers on the first 4 questions, and if she guesses on every question, what is the probability that she will answer every question correctly? ​

Accepted Answer

A) Pr(all correct) =   
B) Pr(all correct) =   
C) Pr(all correct) =   
D) Pr(all correct) =   
E) Pr(all correct) =   
A) Pr(all correct) =   
B) Pr(all correct) =   
C) Pr(all correct) =   
D) Pr(all correct) =   
E) Pr(all correct) =

Question 19

Can the matrix   be a transition matrix?

Accepted Answer

A) Yes 
B) No 
A) Yes 
B) No

Question 20

Suppose a die is biased (unfair) so that each odd-numbered face has probability   of resulting, and each even face has probability   of resulting. Find the probability of getting a number greater than 1. ​

Accepted Answer

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

Compute .

Suppose that a marksman hits the bull's-eye 1 time in 40 shots. If the next 4 shots are independent, find the probability that all of the next 4 shots hit the bull's-eye.

In how many ways can a hand consisting of 4 spades, 3 hearts, 2 clubs, and 1 diamond be selected from a deck of 52 cards?

A bag contains 5 white balls and 5 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn,with replacement, one ball is red R and one is white W.

Forty-eight percent of the U.S. population is female, and 23.3% of the female population have a college degree. If a U.S. resident is chosen at random, what is the probability, to three decimal places, that the person is a female with a college degree?

How many four-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9 if the numbers are less than 9,000 and digits are not used repeatedly? (Hint: Begin with the digit where there is a restriction on the choices.)

A ball is drawn from a bag containing 15 red balls numbered 1-15 and 6 white balls numbered 16-21. What is the probability that the ball is red R and even-numbered E?

A company selling substandard drugs to developing countries sold 2,500,000 capsules with 150,000 of them empty. What is the probability that a person who takes a randomly chosen capsule from this company will get an empty capsule?

An urn contains three red R balls numbered 1, 2 and 3, four white W balls numbered 4, 5, 6 and 7, and three black B balls numbered 8, 9 and 10 A ball is drawn from the urn. What is the probability that it is red and odd-numbered O?

A bag contains 3 nickels, 5 dimes, and 4 quarters. If you draw 3 coins at random from the bag, without replacement, what is the probability that you will get a nickel N, a quarter Q, and a nickel N, in that order?

A box contains 3 red balls, 5 white balls, and 3 black balls. Two balls are drawn at random from the box with replacement of the first before the second is drawn.What is the probability of getting a red R ball on the first draw and a white W ball on the second?

A red ball and 17 white balls are in a box. If two balls are drawn, without replacement, what is the probability of getting 2 red R balls?

Two balls are drawn, without replacement, from a bag containing 18 blue balls numbered 1-18 and 4 green balls numbered 19-22. What is the probability that the first ball is blue and even-numbered and the second ball is even-numbered?

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

An unprepared student must take a 7-question, multiple-choice test that has 5 possible answers per question. If the student can eliminate 2 of the possible answers on the first 4 questions, and if she guesses on every question, what is the probability that she will answer every question correctly?

Can the matrix be a transition matrix?

Suppose a die is biased (unfair) so that each odd-numbered face has probability of resulting, and each even face has probability of resulting. Find the probability of getting a number greater than 1.

Algebraic Concepts

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; Data Description

Derivatives

Applications of Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals: Techniques of Integration

Functions of Two or More Variables

Filters

Exam 8: Introduction to Probability

Compute . ​