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Unformatted text preview: Math 3101 Midterm I October 19, 2010 Instructions: Do all work and record all answers in the blue book, afﬁxing your name and instructor’s
name on front cover. Whenever possible indicate the sample space Q and its cardinality Answers 52) may be left in that form need not be simpliﬁed beyond ﬁnite sums and products; for example ( 5 whereas 2;”:2 3'” should be simpliﬁed. Each problem counts 20 points. 1. A child has ﬁve identical red blocks, four identical green blocks, and one yellow block. Suppose
the child stacks these ten blocks randomly on top of one another (a) In how many distinguishable ways can this be done? (b) Compute the probability that blocks of like color end up stacked together (i.e., the 5 red blocks
are stacked consecutively and the 4 green blocks are stacked consecutively)? 2. (a) In a twenty—question true—false exam, ﬁnd the probability that a. student gets at least 70 percent
correct by guessing. (b) Given that a student gets 7'0 percent correct, what is the probability that the student gets at
least 90 percent? 3. The members of a consulting ﬁrm rent cars from 3 rental companies: 60 percent from Company
A, 30 percent from Company B, and 10 percent from Company C. The past statistics show that 9
percent of the cars from Company A need a tune—up, 20 percent of the cars from Company B need
a tuneup, and 6 percent of the cars from Company C need a tuneup. If a rental car delivered to
the ﬁrm needs a tuneup, what is the probability that it came from Company B? 4. A group of 5 people are interviewed in order to determine the month in which they were born.
Assume for simplicity that all months are equally probable.
Compute the probabilities of the following events:
(a) All 5 people have different birthmonths. (b) One pair of peOple has the same birth—month, a second pair of people has the same birth
month (but a different month from the ﬁrst pair), and the ﬁfth person has a different birth month
from the rest of the group. 5. Consider a succession of rolls of a fair die. (a) Deﬁne the random variable X to be the number of rolls at which 6 appears for the ﬁrst time. i. Compute P(X = 4).
ii. Compute P(X even).
iii. Compute P(X > 5[X even).
(b) Deﬁne the random variable Y to be the number of rolls at which 6 appears for the 4th time.
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 Spring '11
 SARVER

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