M302,
Spring 2005 (Smith)
INFORMATION
FOR THIRD EXAM
Date: Friday, May 6
Time:
Usual class
time (12 noon  12:50 p.m.)
Place:
WEL 1.308 (Same
room as first two exams; see map on information for first exam.)
What you may and may not bring to the exam:
 You
should bring a pencil,
eraser and
scratch paper.
 You
may (and probably should)
bring a nonprogrammable calculator.

You
may not bring notes, your
textbook, or
any programmable device such as a Palm Pilot, cell phone, or graphing
calculator.
Exam procedures:

Please
sit every other seat in the
auditorium.

Cell
phones and pagers should be turned off before the exam begins.
Exam coverage: The
quiz will cover Sections 7.1  7.7, 8.1, and 8.2.
Exam
format: The
quiz will be "semitakehome." By that I mean that the quiz questions
will be very similar to some selection from the following list of
questions. By
"very similar" I mean, for example, that the numbers might be changed
or the wording might be different, or the setting might be different
but the
same techniques can be used. (The retake at the final exam time will be
a
slightly different selection, but also taken from the questions below.)
You
are allowed to get help with these questions before the
exam from
anyone except
me or
the TA or any other math department faculty member or TA . (You
may ask me or the TA questions about anything assigned
for class or covered in class, but not about the questions below or
about very
similar questions, unless they have been assigned in the reading,
assigned as
homework, or discussed in class.) As stated above, you may not bring
any notes
to the exam. Also, be prepared for the questions on the exam to be in a
different order from the corresponding questions below.
Reminder: Unless told otherwise, you
need to explain how
you got your answers to receive full credit.
Posting grades: I will try to get exam grades
posted on Blackboard by 10 p.m., Sunday, May 8.
Possible exam questions:
1. A student was asked to compute a certain
probability. His answer was 2. Could this be correct? Explain.
2. You toss four dice. Find the probability that at
least two of the dice will show the same number.
3. Suppose you have the 26 letters of the alphabet on
separate cards in a hat. Suppose you pick out a card, write down the
letter on it ,
put the card back in the hat, mix up the cards, pick out another card,
and so on. Write an expression for the probability that on your first
five draws your letters (in the order drawn) spell TEXAS.
4. A tetrahedral die is a die in the shape of a
tetrahedron, with the numbers 1, 2, 3, 4 written on the four different
sides. When you toss a tetrahedral die, it lands with one corner up and
one side down, so you look at the number that lands down. Assume all
sides are equally likely to land down.
 What is the probability of rolling a 3 (that is,
of having the 3 land down)?
 Suppose you simultaneously toss two tetrahedral
dice. Make an organized list or chart of all possible outcomes for the
two numbers that land down. Label clearly.
 When you toss two tetrahedral dice, what is the
probability that both numbers landing down are the same? Explain.
 When you toss two tetrahedral dice, what is the
probability that the two numbers landing down have sum less than or
equal to 3? Explain.
5. There are 8 balls in a container. The balls are
labeled 1, 2, 3, 4, 5, 6, 7, 8. For each of the questions a, b, and c
below, tell which of the expressions (i)  (vii) shown here answers the
question, and explain why.
Here are the possible answers:
 8 x 8 x 8 x 8
 (8 x 8 x 8 x 8)/(4 x 4 x 4 x
4)
 (8 x 8 x 8 x 8)/(4 x 3 x 2 x
1)
 (8 x 8 x 8 x 8)/(8 x 7 x 6 x
5)
 8 x 7 x 6 x 5
 (8 x 7 x 6 x 5)/(4 x 4 x 4 x 4)
 (8 x 7 x 6 x 5)/(4 x 3 x 2 x
1)
 None of the above expressions.
Here are the questions:
 You draw out one ball, write down its number, put
it
back in the container, shake up the container, draw out a second ball,
write down its number, put it back in the container, and so on until
you have drawn out 4 balls, putting each one back in the container and
shaking up the container before you draw the next ball. How many
possible lists of four numbers can you get? (For this part, lists with
the same numbers but in different orders are considered different 
for example, getting the numbers 5, 2, 5, 4 would be a different
outcome than getting the numbers 2, 5, 4, 5.)
 You draw out one ball, write down its number, then
draw out the second ball without
replacing the first ball, write down its number, and so on, until you
have four balls. How many different lists of numbers are possible? (For
this part also, lists with the same numbers but in different orders are
considered different  for example, getting the numbers 2, 5, 7, 3
would be a different outcome than getting the numbers 7, 5, 2, 3.)
 You draw out one ball, draw out another ball
(without replacing the first one), and so on until you have four
different balls. You then look at their numbers, not caring about which
number was drawn first. How many different groups of numbers are
possible? (Note that I've said "group" rather than "list," since order
is not taken into account here  so, for example, 2, 5, 7, 3 and 7, 5,
2, 3 would be considered the same groups.)
6. Write down an expression that tells you what the
probability is that at least one person in a room of 50 has their
birthday today. Explain how you got your expression. (Hint: There are
two ways to do this; one is much shorter than the other.)
7. Suppose a University has no regulations about what
courses a student can or cannot take. If it offers 200 courses, write
an expression for the number of different possible combinations of four
different courses a student could take. Explain how you got your
expression.
8. A roulette wheel has 36 spaces numbered 1 to 36,
with half of them red and half of them black. There are also two green
spaces, labeled 0 and 00. There is little slot by each space, and a
little ball that can fit into the slot. When the wheel is spun, the
ball goes around and lands in one of the slots when the wheel stops.
There are various kinds of bets that can be made.
 What is the probability that the ball lands on 15?
Explain.
 What is the probability that the ball lands on a
red space? Explain.
 What is the probability that the ball lands on an
odd number? Explain.
 What is the probability that the ball lands on a
number from 1 to 12? Explain.
 If you bet $10 on odd, you get $20 back if the
ball lands on an odd number.
 What is the expected value of this bet? Show
your steps.
 What will the casino expect to make off of 100
bets like this?
 What payoff would be needed to make this game
fair?
f. If you bet $10 that the ball would
land on a
number from 1 to 12, you will get back $30 if the ball does land on one
of these numbers from 1 to 12.
 What is the expected value of this bet? Show
your steps.
 What will the casino expect to make off of 100
bets like this?
 What payoff would be needed to make this game
fair?
9. Suppose you play a game consisting of someone
handing you a dime and a quarter. You flip the dime and quarter and
keep all the coins that come up heads. What is the expected value of
that game? Show our work. [Hint: It should help to make an organized
list of all possible outcomes and what you win or lose for each.]
10. You play the following game with a friend: You
flip two coins. If they both come up heads, your friend gets a dollar.
Otherwise, you keep the money he bet. How much should your friend bet
so that the game is fair? Explain.
11. If you asked each student in a class how many
pets they had ever had (in their entire life) and then made a graph
(histogram) of the information you obtained, which of the following
would it most resemble? Explain why.
a
b
c
d
If the histogram
doesn't print, try this pdf file.
12. The criminal justice system wants to know about
repeat offenders, so they gather 1000 excons who were released two
years ago and not arrested since then. They want to know how many
excons have committed felonies since their release. Of course, none of
those present would want to answer such a question. So the authorities
ask everyone in the room to secretly flip a coin. Then they ask people
to raise their hands If they flipped a head OR committed a felony. 620
people raise their hands. Estimate how many of the 1000 excons
committed felonies since their release. Clearly show all work leading
to your conclusion.
13. All athletes participating in a regional high
school track and field championship must provide a urine sample for a
drug test. Those who fail are eliminated from the meet and suspended
from competition for the following year. Studies show that, at the
laboratory that analyzes the urine samples, the drug tests are 95%
accurate. That means that 95% of the time when the athlete has been
using drugs, the test says "positive," and similarly, 95% of the time
when the athlete has not been using drugs, the test says "negative." If
only 40 out of the 1000 athletes at the meet actually use drugs, what
fraction of the athletes who fail the test (i.e., get a positive test
result) are falsely accused and therefore suspended even though they
have not used drugs?
14. Discuss the validity of the reasoning in each of
the following situations. Tell whether the reasoning is sound or not,
and why.
a. Sales of lemonade in a restaurant are positively
correlated with ticket sales at the swimming pool a block away.
Therefore increased use of the swimming pool causes an increase in the
purchase of lemonade.
b. A report
by the U.S.
Commissioner of Narcotics on a study of 2,213 hardcore narcotic addicts
in
the Lexington, Kentucky, Federal Hospital showed that 70.4% smoked
marijuana
before using heroin. This shows that marijuana use causes people to use
heroin.
15. There are 100
students in a class. One
student claims that the probability that at least two
students in the class have their birthday in the same week is 1.
Is she right or wrong? Explain.
16. You flip a penny and dime.
a. What is the probability of getting both heads?
b. What is the probability that there is at least
one head?
c. You can't see the outcome, but are told that one
of the coins (you don't know which) shows a head. What is the
probability that both are heads?
d. You
can't see the outcome, but are told that the penny shows heads. What is
the probability that both are heads?
17. The students in a class were asked how many music CD's they own.
The following graph shows the data collected:
X
X
X
X
X X
X
X
X
X X
X
X
X X
X
X
X X
X
X
X
X
X
049 5099 100149 150199
200249 250299 300349 350  399 400449 450  499
Would the mean or the median better describe the typical number of
music CD's a student in the class owns? Why? How would the mean compare
in this example? Why?
18. A recent poll claims that with a margin of error of 5%, 47% of
American's eat broccoli. The next week, the poll shows that 51% eat
broccoli. Should broccoli growers be encouraged? Explain.
19. Here are the 1983 mileage (Miles per Gallon) figures for European
and Japanese cars available in the U.S.:
Peugeot604SL 14
Plym.Arrow 28
Plym.Champ 34
Plym.Horizon 25
Plym.Sapporo 26
Plym.Volare 18
Pont.Catalina 18
Pont.Firebird 18
Pont.GrandPrix 19
Pont.LeMans 19
Pont.Phoenix 19
Pont.Sunbird 24
RenaultLeCar 26
Subaru
35
ToyotaCecila 18
ToyotaCorolla 31
ToyotaCorona 18
VWRabbit 25
VWRabbitDiesel 41
VWScirocco 25
VWDasher 23
Volvo260
17
a. Make a histogram of the mileages. Use intervals
1115, 1620, 2125, 2630, 3135, 36  40, 41  45.
b. Based on your histogram, how would you describe
the distribution of the mileages?
c. Make a fivenumber summary of the data.
20. Click here to see two normal
distributions. Which one has the larger standard deviation? How do you
know?