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HOMEWORK: MATH 304
Homework 0: Due: Friday, 20 January 2017
Briefly relate (in one or two paragraphs) information about yourself that will help me get
to know you. If you wish, you may let the following questions serve as a guide: When
did you take Math 161 and 162 (or their equivalents)?; why are you taking Math 263
now? (for example: "major requirement", "just for fun because I love mathematics",
"nothing else fits my schedule", "my parents forced me to take this course", "I am looking
for an easy A to raise my gpa", “I want to understand Maxwell’s equations”); what is
your major?; what is your career goal?; what has been the nature of your previous
experience with math either in high school or in college (that is, have you enjoyed math in the past? Do you
like to see applications more than theory, or do you prefer theory?).
(Post your response in Piazza as a private note in the folder hw0.)
Homework I: Due: Tuesday, 24 January 2017
Read Chapter 1 paying particular attention to sections 1 and 2.
Highly recommended: Watch MIT 6.041 lecture 1 (probability models and axioms.
I would like you to come prepared on Thursday knowing the definitions of sample space and event as well as
the basic set operations (union, intersection, complement). Also, learn the basic properties of these operations
as well as deMorgan’s rules.
Submit solutions to exercises 1, 5, 6, and 8 of chapter 1.
Begin by stating the problem that you are solving. Submit (clearly and neatly written) solutions to the
assigned exercises. Loose pages will not be accepted; they must be stapled!
Finally, if you discover websites that you particularly like, please share them by posting the URLs in piazza.
Homework II: Due: Tuesday, 31 January 2017
Read section 1.7 of our text. Watch Lecture 4 (Counting)
1. Albertine, a fly, lives at vertex A of a cube and is imprisoned in her home. A
spider, Charlotte, lives at the opposite vertex B. Charlotte is quite mobile. She
would like to reach A in five or fewer steps. (Here a step means traveling from
one vertex to an adjacent one.) Charlotte will stop as soon as she reaches
Albertine. In how many ways can Charlotte achieve her objective?
2. From a drawer that contains 13 pairs of gloves (each pair of a different style and color), six gloves
are randomly selected. Find the probability of obtaining exactly two pairs of gloves of the same style
and color. (You need not simplify your answer.) Begin by defining an appropriate equiprobable
sample space.)
3. Little Red Riding Hood lives in the far northwest; her grandmother lives in the far southeast. How
many distinct paths are there from Little Red Riding Hood’s dwelling to that of her grandmother if
she can move only from one adjacent * to the next either right, down, or diagonally down to the right
(at 45°).
4. Give a story proof of Vandermonde’s identity, viz.
 m  n k  m  n

     
k

 j 0  j   k 


j 
Homework III: Due: Tuesday, 7 February 2017
Review chapter 1. Read sections 2.1 and 2.2. Watch Lecture 2 (Conditioning & Bayes Theorem)
1. Urn A contains 2 white and 4 red balls, whereas Urn B contains 1 white and 1 red
ball. A ball is randomly chosen from Urn A and put into Urn B, and a ball is then
randomly selected from Urn B. What is
(a)
the probability that the selected ball from Urn B is white?
(b) the conditional probability that the transferred ball was white given that a white
ball is selected from Urn B?
2.
Vladimir recently earned a B.S. from Loyola University Chicago. Instead of
waiting, he decides to be proactive by taking the first three actuarial exams next
summer. He will take the first exam in June. If he passes that exam, then he will take
the second exam in July, and if he also passes that one, then he will take the third
exam in September. If he fails an exam, then he is not allowed to take any others. The probability that he
passes the 1st exam is 0.9. If he passes the 1st exam, then the conditional probability that he passes the 2nd
exam is 0.8. If he passes both the 1st and the 2nd exams, then the conditional probability that he passes the
3rd exam is 0.7.
(a)
What is the probability that Vladimir passes all 3 exams?
(b) Given that Vladimir did not pass all 3 exams, what is the conditional probability that he failed the
2nd exam?
3. Assume that each component has success probability of p and that the components are independent of each
other.
(a) Compute the probability of success for the given system.
(b) Find the conditional probability the system works given that at least one of the 3 components in
parallel fails.
4. Swann throws a three-sided die with faces numbered 1, 2, and 3. Assume that the probability of 1
appearing is ½; the probability of 2 appearing is ¼; and the probability of a 3 appearing is ¼. Consider a
sequence of six independent rolls of this die.
(a) What is the probability that exactly three of the rolls have result equal to 3?
(b) What is the probability that the first roll is 1, given that exactly two of the six rolls have result of
1?
(c) We are told that exactly three of the rolls resulted in 1 and exactly three resulted in 2. Given this
information, what is the probability that the sequence of rolls is 121212?
Homework IV: Due: Tuesday, 14 February 2017
Read sections 2.1, 2.2, 2.3 and 2.4. Watch Lecture 5 (discrete random variables, pmfs, expectation)
All knowledge resolves itself into probability.
- David Hume
There are times when truth hardly seems probable.
- Nikolas Boileau
Lest men suspect your tale untrue, keep probability in view.
- John Gay
The theory of probabilities is at bottom nothing but common sense reduced to calculus.
- Pierre Simon de Laplace
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