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MATH 355 FIRST MIDTERM TAKEHOME TEST SPRING 2009
NAME _______________________________ cwid___________________________________
You should turn in your exam no later than 11:05 a.m. February 9, 2009
No exam will be accepted after the deadline.
This is an open book and class note exam. You may only use text book, class note, and calculator.
You must turn in a signed copy of the honor statement below.
………………………………………………………………………………………………………………………………………………………………
HONOR STATEMENT
I HEREBY CONFIRM THAT I DID MY OWN WORK ON THIS EXAM.
I did not consult or get help from any other person, source, or internet.
Name (print) ________________________________________
Date _______________________
Signature ____________________________________________
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1. (4 pts) Suppose that A and B are two events. Write expressions involving unions, intersections,
and complements that describe the following:
(a) Both events occur.
(b) At least one occurs.
(c) Neither occurs.
(d) Exactly one occurs.
2. (4 pts) Suppose two dice are tossed and the numbers on the upper faces are observed. Let S
denote the set of all possible pairs that can be observed. Define the following subsets of S:
A = event that the number on the second die is even.
B = event that the sum of the two numbers is seven.
C = event that at least one number in the pair is odd.
List the elements in A, C, AB, AB, A B and AC.
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3. (4 pts) A survey classified a large number of adults according to whether they were diagnosed as
needing eyeglasses to correct their reading vision and whether they use eyeglasses when
reading. The proportions falling into four resulting categories are given in the table.
Needs Glassed
Yes
No
Uses Eyeglasses for reading
Yes
No
.44
.14
.02
.40
If a single adult is selected from the large group, find the probabilities of the events defined
below.
(a) The adult needs glasses.
(b) The adult needs glasses but does not use them.
(c) The adult uses glasses whether the glasses are needed or not.
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4. (5 pts) A business office orders paper supplies from one of three vendors, V , V , or V . Orders
are randomly to be placed to those three vendors on two successive days, one order per day.
Thus (V , V ) might denote that vendor V gets the order on the first day and vendor V gets
the order on the second day.
(a) Describe the sample space of this experiment in ordered pairs on two successive days.
(b) Assume the vendors are selected at random to fill the order. Assign a probability to each
elementary outcome.
(c) Let A denote the event that the same vendor gets both orders and B the event that the
second vendor, V ,gets at least one order. Find P(A), P(B), P(A B), and P(A B).
5. (4 pts) An airline has six flights from Birmingham to Los Angles and seven flights from Los Angles
to Hawaii per day. If the flights are to be on made separate days, how many different flight
arrangements can the airline offer from Birmingham to Hawaii.
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6.
(4 pts) How many different ten-digit telephone numbers can be formed if the first digit and the
4th digit cannot be zero?
7.
(5 pts) An unbiased die is tossed six times and the number on the uppermost face is recorded
each time. What is the probability that the numbers recorded are 1, 2, 3, 4, 5, and 6 in any
order?
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8. (5 pts) A study is to be conducted in a hospital to determine the attitudes of nurses toward
various administrative procedures. There are 90 nurses work for this hospital and 20 of them are
male. A random sample of 10 nurses to be selected. What is the probability that the sample of
ten will include exactly 4 male and 6 female nurses?
9. (6 pts) A fraternity is conducting a raffle where 50 tickets are to be sold – one per person. There
are three prizes to be awarded. If you and three friend each buy one ticket. What is the
probability that you and your (three) friends
(a) win all of the prizes?
(b) win exactly two of the three prizes?
(c) win exactly one of the prizes?
(d) win none of the prizes?
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10. (6 pts) A labor dispute has arisen concerning the distribution of 20 laborers to four different
construction jobs. The first job (considered to be very undesirable) required 6 laborers; the
second, third, and fourth utilized 4, 5, and 5 laborers, respectively. The dispute arose over an
alleged random distribution of the laborers to the jobs which placed all four members of a
particular ethnic group on job 1. In considering whether the assignment represented injustice, a
mediation panel chaired by Dr. Hsia who hired you to find the probability of the disputed event
under the assumption that the laborers are randomly assigned to jobs. Give Dr. Hsia your report.
11. (5 pts) If two events, A and B, are such that P(A) = 0.5, P(B) = 0.3, and P(AB) = 0.1, find the
following:
(a) P(A B)
(b) P(B A)
(c) P(A A B)
(d) P(A AB)
(e) P(AB A B)
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12. (7 pts) A study of the post-treatment behavior of a large number of drug abusers suggests that
the likelihood of conviction within a 2-year period after treatment may depend upon the
offender’s education. The proportions of the total number of cases falling in four educationconviction categories in the following table.
Status Within Two Years After Treatment
Education
Convicted
Not Convicted
10 years or more
.10
.30
9 years or less
.27
.33
Totals
.37
.63
Total
.40
.60
1.00
Suppose that a single offender is selected from the treatment program. Define the events:
A: The offender has 10 or more years of education.
B: The offender is convicted within 2 years after completion of treatment. Find
(a) P(A)
(b) P(B)
(c) P(AB)
(d) P(A B)
(e) P(A B)
(f) P(A B)
(g) P(B A)
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13. (5 pts) A smoke detector system uses two devices, A and B. If smoke is present, the probability
that it will be detected by device A is 0.95; by device B, 0.90; and by both devices, 0.88.
(a) If smoke is present, find the probability that the smoke will be detected by at least one of
those two devices.
(b) Find the probability that the smoke will be undetected.
14. (4 pts) Five radar sets, operating independently, are set to detect any aircraft flying through a
certain area. Each set has a probability of .05 of failing to detect a plane in its area.
(a) If an aircraft enters the area, what is the probability that it goes undetected?
(b) If an aircraft enters the area, what is the probability that it is detected by all five radar sets?
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15. (8 pts) Suppose that two defective refrigerators have been included in a shipment of six
refrigerators. The buyers begin to test the six refrigerators one at a time.
(a) What is the probability that the last defective refrigerator is found on the fourth test?
(b) What is the probability that no more than four refrigerators need to be tested to locate both
of the defective refrigerators?
(c) When given that exactly one of the two defective refrigerators has been located in the first
two tests, what is the (conditional) probability that the remaining defective refrigerator is found
in the third or fourth test?
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16. (8 pts) A plane is missing and is presumed to have equal probability of going down in any of
three regions. If a plane is actually down in region i (I = 1,2,3), let .6, .7, .8 denote probabilities
that the plane will be found upon a search of the ith region, i = 1,2,3, respectively.
(a) What is the conditional probability that the plane is in region 1 given that the search of
region 1 was unsuccessful?
(b) What is the conditional probability that plane is in region 2 given that the search of region 1
was unsuccessful?
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17. (8 pts) The National Highway Traffic Safety Administration is interested in the effect of seat belt
use on saving lives. The following table shows a report on children under the age 5 who did not
wear seat belt and were involved in motor vehicle accidents in which at least one fatality
occurred between 1985 and 1989.
Ages
0
1
2
3
4
Survival Statistics
Survivors
Fatalities
45%
55%
64%
36%
71%
29%
75%
25%
77%
23%
Age % in report
14%
16%
23%
22%
25%
Find the conditional probability that a child is 3-year old given that he survived in an accident.
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18. (8 pts) Suppose that two unbiased dice are tossed repeatedly and the sum of the two
uppermost faces is determined on each toss. What is the probability that we obtain a sum of 3
before we obtain a sum of 7?
(HINT: define events A = obtain a sum of 3; B = do not obtain a sum of 3 or 7)