Review for Exam 2
MA 111
Fall 2006
Exam 2 is on Tuesday, October 31. We will review for the exam in class on Thursday,
October 26. I hope to schedule review sessions out of class later this week and maybe on
Monday of next week. I might hand out a practice exam on Thursday.
You should review your homework and class notes. Good review problems for the exam
are on pages 146-149 and pages 205-210 of the book.
You should know the following topics for the exam:
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How to interpret Venn diagrams and use them to count the numbers of elements
in various sets. How to use tables to count the numbers of elements in various
sets (Section 3.3).
How to use the Basic Counting Law, permutations, and combinations to solve
various counting problems (Section 3.4).
What probability is. The difference between theoretical and experimental
probabilities. Know the Law of Large Numbers (Section 4.1).
How to find probabilities of events by using counting techniques (Sections 4.14.2).
What the odds of an event are. Know how to convert from odds to probabilities
and vice-versa. Know what house odds means, and know how to determine if a
bet is fair (Section 4.3).
How to compute probabilities of compound events using tables, formulas, Venn
diagrams, and tree diagrams (Section 4.4).
How to compute conditional probabilities using tables, formulas, Venn diagrams,
and tree diagrams (Section 4.5).
What expected value is and how to compute it (Section 4.6).
Here are some practice problems for you to work on. Note that not all types of problems
that will appear on the exam are listed below.
1. In a college of 1000 students, 495 do not use tobacco. Out of the students that do use
tobacco, 385 using smoking tobacco (cigarettes, pipes, cigars), and 150 use nonsmoking
tobacco (dipping tobacco or chewing tobacco or snuff). How many students use both
smoking tobacco and nonsmoking tobacco?
2. Suppose three brands—Brand A, Brand B, and Brand C—of some product exist. Out
of 800 people, 35 have tried all three brands. Furthermore, 100 have tried Brands A and
B, 90 have tried B and C, and 115 have tried A and C. Finally, 300 have tried Brand A,
270 have tried Brand B, and 250 have tried Brand C. Draw a Venn diagram and
determine the number in each region. Explain what each number means. There should
be a total of 8 regions.
3. Solve the following counting problems. Clearly indicate your methods.
a. You are dealt 6 cards. How many such hands have exactly three aces? (The
other three cards may have a pair or a three of a kind among them.) How many
have any three of a kind?
b. A contest involves guessing which five balls from this special vat have been
pulled. Order does not matter. The special vat consists of 10 numbered balls (09) of each color. The colors are red, green, blue, white, yellow, and black. In
how many ways can exactly four balls of the same number be chosen? In how
many ways can exactly four balls of the same color be chosen?
c. Three regular dice are thrown. In how many ways can a sum of 12 be
obtained?
d. I am asked to choose a color for a large shirt, a color for a medium shirt, and a
color for a small shirt. I have 40 colors to choose from, and I may not repeat
colors. In how many ways may I do this?
4. Find the probability of each of the following events.
a. Find the probability of getting such a hand described in 3a above.
b. Find the probability of getting four balls of the same number, as described in
3b above. Find the probability of getting exactly four balls of the same color.
How about five balls of the same color?
c. A hundred index cards are numbered 1 to 100, and I draw 7 cards without
replacement. What is the probability that at least one of these cards is an even
number?
5. The probability of a bet succeeding is 1/17. The casino says that the odds of the bet
succeeding are 14:1. Is the casino correct or not? If the casino is not correct, then what
odds should the casino post?
6. Find each probability. Use either a Venn diagram, a formula, or a tree diagram to help
you.
a. Out of a group of 267 people, 99 people support UT and 136 people support
UK. However, 46 do not support either school. What is the probability that a
person chosen at random supports both schools? What is the probability that a
person supports exactly one of the schools?
b. Fred challenges Josh to an arm wrestling contest. They play three games.
Fred has a 65% chance of winning any game. What is the probability that Fred
will win at least two games? What is the probability that Josh will win at least
one game?
c. Four cards are chosen randomly without replacement from a deck of cards.
What is the probability that at least one ace is drawn in the last two cards if the
first two cards were aces?
d. Suppose three dice are tossed. What is the probability that the sum of the dice
is 14 if two of the dice are odd numbers? What if exactly one die is an odd
number?
7. I toss two dice. The sum is recorded, and I win money according to the following
scheme:
If the sum is either 2, 3, 4, or 5—I lose $1.
If the sum is 6, I win $2.
If the sum is 7, I lose $2.
If the sum is 8, 9, or 10—I win $1.
If the sum is 11, I win $3.
If the sum is 12, I win $4.
What is the expected value of this game? Is the game fair? Why or why not? Would it
be a good idea for me to play this game?
8. A lottery involves picking 5 winning numbers from a set of 36 possible numbers. It
costs $2 to buy a ticket. If the ticket has none or 1 or 2 of the winning numbers correct,
then the lottery pays the player nothing. If the ticket has exactly three of the winning
numbers correct, then the lottery pays $30. If the ticket has exactly four of the winning
numbers correct, then the lottery pays $1500. Finally, if the ticket has all five winning
numbers correct, then the lottery pays $100,000. What is the expected value of this game
(include the cost to play)? Interpret this number. Is the game fair? Explain. If the game
is not fair, then how much should the lottery pay for the ticket that has all 5 winning
numbers correct to make this a fair game? (Keep the payouts for the other cases the
same.)