Download Practice Problems for Midterm Exam I

Practice Problems for Midterm Exam I
Problem 1. Given the following Venn Diagram, along with some probabilities.
P (e2 ) = P (e4 ) = 0.12,
P (e3 ) = 3P (e2 ),
and P (A) = 0.20.
(a) Find P (e1 ) and P (e5 ).
(b) Give composition and probabilities for B C and A ∪ C.
(c) Find P (C|B).
(d) Are the events A and B independent? Why or why not?
Problem 2. In the 1st stage of an experiment a fair die will be rolled and in the 2nd stage a
fair coin will be tossed. Use 1, 2, 3, 4, 5, 6 for die and H, T for coin.
(a) Draw a tree diagram and find the sample space.
(b) Suppose A = {odd number on die} and B = {tail on coin}. Assuming all elementary
outcomes are equally likely, find P (A), P (B), P (AB) and P (A ∪ B).
(c) Are A and B independent? Give calculations in support of your answer.
Problem 3. Diane goes to Bruegger’s and purchases a baker’s dozen (thirteen) bagels. A half
dozen are plain, five are blueberry, and the other two are cinnamon raisin.
(a) Suppose that Diane blindly grabs two bagels for lunch. What is the probability that she
will take two plain bagels?
(b) Suppose that Diane blindly grabs three bagels. What is the probability that she will take
exactly one blueberry bagel?
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(c) Suppose that Diane packs one bagel for each of her three children. What is the probability
that all the children get different types of bagels? What is the probability that all the
children get cinnamon raisin bagels?
Problem 4. Suppose that ninety percent of U.S. high school graduates have attended public
schools. I will randomly ask three high school graduates whether they have attended a public
school or not. Assume responses are independent.
(a) Construct a tree diagram to represent the responses I get, and list the sample space. (Use
P=public, N=non-public)
(b) Find the probability that exactly two people surveyed attended public school.
(c) Find the probability that at least two people surveyed did not attend public school.
Problem 5. The management of a grocery store are looking at 500 purchases over one
day. They are interested in two features, A = {purchase is of amount greater than $50} and
B = {purchase is paid by cash (including cheque)}. Out of 500 purchases 60 were of amount
greater than $50, 65 were paid by credit card, and 20 were of amount greater than $50 but paid
by cash.
(a) From the above information construct a two-way table.
(b) Find P (A B), P (A B).
(c) Find P (B|A) and P (B|A).
(d) Are A and B independent? Why or why not?
Problem 6. 15 persons who donated blood at a Red Cross center on one day were typed for
blood and the following counts were found.
Blood Group
No. of persons
O
3
A
5
B
6
AB
1
Total
15
Two of those persons later developed flu but which two were not known. To test for virus two
blood samples would be selected at random (without replacement) from the 15 samples above.
(a) Find the probability that both have blood type B.
(b) Find the probability that one has blood type A and the other different from A.
(c) Find the probability that none has blood type O.
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Problem 7. Suppose that we have a box with 20 marbles which are either blue or yellow.
Also we know that each marble can be either small or large. We are told that 8 marbles are
blue, 15 marbles are large, and of the 15 large marbles 9 are yellow. One marble is chosen at
random.
(a) What is the probability that the selected marble will be blue and large?
(b) Given that the selected marble will be yellow what is the (conditional) probability that
the marble will be small?
(c) Let B be the event that the selected marble will be blue, and S be the event that the
selected marble will be small. Are the events B and S independent?
Problem 8. A quiz consists of two questions. Each question has three possible answers, true
(T), false (F), and incomplete information (I), of which only one is correct. A student arrives
unprepared and answers the questions by selecting one answer out of three at random for each
question. Suppose X is the number of questions correctly answered.
(a) What are the possible values of X?
(b) For each possible value of X find its probability.
(c) Give the probability distribution of X.
Problem 9. Bob, John and Linda went to dine at a restaurant. They agreed to the following
rule to decide who would pay the bill: each of them would toss a coin and the one who obtained
a result different from the other two would pay the entire bill. If all of them got the same result
the bill would ge shared equally by them.
(a) Let X denote the random variable “Bob’s contribution towards paying the bill”. If the
total bill was $60 then find the probability distribution of X.
(b) What is the probability that the bill would be equally shared by them?
Problem 10. The University Book Store received a late shipment of 25 statistics books from
the publisher. Packaged with each of the books is a copy of the statistical software that is being
used in the statistics classes. Unknown to the store is that five of the disks shipped contained
an error that will not allow the student access to the software. Suppose you and your friend
both purchase books from the late shipment of 25.
(a) Find the probability that only you got a disk with the error.
(b) Find the probability that at least one of your disks contains the error.
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Problem 11. In a summer session two sections of Stat 11 are generally offered. If the number
of registered students in a section is less than 20 it is combined with the other section and offered
as just one section. The probability that a section will have less than 20 registered students is
0.4 and the sections are formed independently. Find the probability that both sections would
be combined.
Problem 12. Suppose you are given a random variable X with the following probability
distribution
x
1
2
3
4
5
f (x)
.35
.2
.15
(a) You are told that P (X = 5) = 2 · P (X = 3). Using this information fill up the table
above.
(b) Find P (X > 2), P (X ≥ 1) and P (X > 5).
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