MATH 1070-070: Quiz 3
June 5, 2008
Name:
No outside materials are allowed except pens, pencils, erasers, and calculators. You have one
half hour to complete this quiz. Please keep nervous ticks to a minimal so as to not disrupt
anyone else taking the quiz. Anyone caught cheating will be punished with a grade of 0%.
You may need the following:
µ=
n
X
xi P (xi )
i=1
c
P (A ) = 1 − P (A)
P (A or B) = P (A) + P (B) − P (A and B),
and, if A and B are independent
P (A and B) = P (A)P (B)
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1. (Chapter 5) After the major accounting scandals with Enron, a large energy comapany, the question
may be posed, “Was there any way to examine Enron’s accounting books to determine if they had been
‘doctored’ ?” One way uses Benford’s Law. This states that in a variety of circumstances, numbers
as varied as populations of small towns, figures in a newspaper or magazine, and tax returns and
othere business records begin with the digit “1” more often than other digits. This law states that the
probabilities for the digits 1 through 9 are approximately:
Digit
Probability
1
0.3
2
0.18
3
0.12
4
0.10
5
0.08
6
0.07
7
0.06
8
0.05
9
0.04
(a) If we were to randomly pick one of the digits between 1 and 9 using a random number table (and
not the list of probabilities above), what is the probability for each digit?
(b) When people attempt to fake numbers, there’s a tendency to use 5 or 6 as the initial digit more
often than predicted by Benford’s law. What is the probability of 5 or 6 as the first digit by (i)
Benford’s law, (ii) random number selection?
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2. Three fifteen-year old friends with no particular background in driver’s education decide to take the
written part of the Georgia Driver’s Exam. Each exam was graded as a pass (P) or a failure (F).
(a) How many outcomes are possible for the grades received by the three friends together? Using a
tree diagram, list the sample space.
(b) If the outcomes in the sample space in (a) are equally likely, find the probability that all three
pass the exam.
(c) In practice, the outcomes in (a) are not equally likely. Suppose that statewide 70% of fifteen-year
olds pass the exam. If these three friends were a random sample of their age group, find the probability
that all three pass.
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3. Let X = number of languages in which a person is fluent. According to Statistics Canada, for residents
of Canada this has probability distribution P (0) = 0.02, P (1) = 0.81, P (2) = 0.17, with negligible
probability for higher values of x.
(a) Explain why it does not make sense to compute the mean of this probability distribution as
(0 + 1 + 2) /3 = 1.
(b) Find the correct mean.
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