Math 105Review Problems for Exam on Counting and Probability
(Chapters 12 and 13)
1. Review all in-class examples. Recall that it is very important to use proper probability notation for ALL
problems involving probabilities. You WILL LOSE POINTS if you do not use the proper notation.
2. A group consists of 5 women and 7 men.
a. In how many ways can a 3-member committee be chosen from the group?
b. In how many ways can a 3-member committee be chosen if the committee must contain at least one
woman and at least one man?
c. In how many ways can a president, vice president, and treasurer be chosen from the group?
3. In how many ways can five people line up at a bank teller’s window?
4. A jar contains four red and six white balls. Three balls are drawn at random. In how many ways can at
least one red ball be drawn?
5. You flip a coin 10 times. In how many ways could you obtain at least two heads?
6. A test consists of 10 questions.
a. In how many ways can the test be answered if the possible answers are “true” and “false?”
b. In how many ways can the test be answered if the possible answers are “true,” “false,” and
“maybe?”
c. In how many ways can the test be answered if the possible answers are (a), (b), (c), (d), or (e)?
7. A test consists of 4 true/false questions.
a. If the questions are answered randomly, what is the probability of answering all questions correctly?
b. If the questions are answered randomly, what is the probability of answering 2 or more questions
correctly?
c. If the questions are answered randomly, what is the probability of answering exactly 3 questions
correctly?
8. Suppose license plates in a certain state have 4 digits followed by 2 letters.
a. How many possible plates exist?
b. How many plates exist if digits cannot be repeated but letters may be repeated?
c. How many plates exist if neither digits nor letters can be repeated?
d. What is the probability of choosing a random plate that has no digits nor letters repeated?
e. What is the probability of choosing a random plate that has the letters AB (in any order)?
9. Suppose a single card is selected from an ordinary deck of cards. Use a Venn diagram to represent each
situation and then determine the probability of choosing a card that is as described. Leave each answer
as a simplified fraction.
a. a five or a club.
b. a five and a club.
c. a queen or a king.
d. a queen and a king.
e. a face card or a diamond
f. a face card and a diamond
10. Suppose that you toss a coin and roll a die.
a. Determine the sample space of the event.
b. What is the probability of rolling an even number?
c. What is the probability of rolling an even number and tossing heads?
d. What is the probability of rolling a 5 or tossing heads?
e. What is the probability of rolling a 5 and tossing heads?
f. What is the probability of rolling 1, 2, 3, or 4?
11. Do problem numbers 4, 5, 6, and 7 in section 13.1.
12. Suppose that you roll two dice. You will be paid $5 if you roll a double. You will not receive anything
for any other outcome. How much should you be willing to pay for the privilege of rolling the dice?
13. A box contains one each of $1, $5, $10, $20, $50, and $100 bills. It costs $20 to reach in and withdraw
one bill. What is the expected value?
14. A game involves tossing two coins and receiving 50¢ if they are both heads. What is a fair price to pay
for the privilege of playing?
15. A game at a fund-raiser carnival involves throwing a dart at a wall of balloons. Hitting a balloon of a
certain color determines your prize. To play, you must pay a non-returnable fee of $5.00. Suppose that
52% of the balloons are red, 26% are green, 14% are purple, 6% are blue, and 2% are yellow. The
following table provides the player with a list of the prizes:
Color
Red
Green
Purple
Blue
Yellow
Prize
$1.00
$2.00
$10.00
$15.00
$25.00
a. What is the expected value of the game? Your answer should be in dollars and cents.
b. Briefly explain what this expected value means about playing the game. Be specific.
16. Do problem number 35 in section 13.2.
17. What are the odds in favor of drawing an ace from an ordinary deck of cards?
18. The probability of drawing a heart from a deck of cards is ¼; what are the odds in favor of drawing a
heart?
19. Suppose the probability of an event occurring is 0.82. What are the odds in favor of this event?
20. The odds against an event are ten to one. What is the probability of this event?
21. A sorority has 35 members, 25 of whom are full members and 10 are pledges. Two people are selected
at random from the membership list of the sorority. Find the requested probabilities.
a.
b.
c.
d.
e.
f.
g.
The first person selected is a pledge.
The first person selected is not a pledge.
The second person selected is a pledge if the first person selected was a pledge.
The second person selected is a full member if the first person selected was a full member.
The second person selected is a pledge if the first person selected was a full member.
The second person selected is a full member if the first person selected was a pledge.
The second person selected is a pledge.
22. Suppose a die is rolled twice and let
A = {first toss is a prime}
C = {second toss is a 2}
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
Are A and B independent?
Are A and C independent?
Are A and D independent?
Are B and C independent?
Are B and D independent?
Are C and D independent?
P A  B 
P A  C  
P A  B 
P A  C  
PB  D 
PC  D 
B = {first toss is a 3}
D = {second toss is a 3}