Research and Lecture notes

Exercise 1. Two dice are simultaneously rolled. For each pair of events defined below, compute if they are independent or not.

(a) A1 ={thesumis7},B1 ={thefirstdielandsa3}.

(b) A2 = {the sum is 9}, B2 = {the second die lands a 3}.

(c) A3 = {the sum is 9}, B3 = {the first die lands even}.
(d) A4 = {the sum is 9}, B4 = {the first die is less than the second}.

(e) A5 = {two dice are equal}, B5 = {the sum is 8}.
(f) A6 = {two dice are equal}, B6 = {the first die lands even}.

(g) A7 = {two dice are not equal}, B7 = {the first die is less than the second}.

Exercise 2. Are the events A1, B1 and B3 from Exercise 1 independent?

Exercise 3. Suppose you toss a fair coin repeatedly and independently. If it comes up heads, you win a dollar, and if it comes up tails, you lose a dollar. Suppose you start with $20. What is the probability you will get to $150 before you go broke?

Exercise 1. Suppose that A and B are pairwise disjoint events for which P(A) = 0.3 and P(B) = 0.5.

1.   What is the probability that B occurs but A does not?
2.   What is the probability that neither A nor B occurs?

Exercise 2. Forty percent of the students at a certain college are members neither of an academic club nor a Greek organization. Fifty percent are members of an academic club and thirty percent are members of a Greek organization. What is the probability that a randomly chosen student is

1. Suppose a License plate must consist of a combination of 8 numbers or letters. How many license plates are there if:
   1. there can only be letters?
   2.  the first three places are numbers and the last five are letters?
   3. the first four places are numbers and the last four are letters, but there can not be any repetitions in the same license plate?
2.  A school of 60 students has awards for the top math, English, history and science student in the school
   1. How many ways can these awards be given if each student can only win one award?
   2. How many ways can these awards be given if students can win multiple awards?
3.   An iPhone password can be made up of any 6 digit combination.
   1. How many different passwords are possible?
   2. How many are possible if all the digits are odd?
4. Suppose you are organizing your textbooks on a book shelf. You have three chemistry books, 5 math books, 5 history books and 5 English books.
   1. How many ways can you order the textbooks if you must have math books first, English books second, chemistry third, and history fourth?
   2. How many ways can you order the books if each subject must be ordered together?

The main educational resource for MA3160 is the following webpage: UConn Undergraduate Probability OER.

No book is required and the course will mostly be based on the lecture notes posted here.

There will be two midterm exams (in class) and a final exam whose dates will be communicated later.

There will be weekly homework assignments (not graded).  Each Thursday, at the end of the class, there will be a 15 minutes quiz consisting of one or two of the homework problems picked randomly.

The final grade will be made of 20% first midterm, 20% second midterm, 20% quiz and 40% final exam.

The following topics will be covered.

The Department of Mathematics at the University of Connecticut is designating Fall 2018 as a Special Semester in Probability.

Special lectures:

Srinivasa Varadhan, 9/13
Ofer Zeitouni, 9/20
Elizabeth Meckes, 9/27
Walter Schachermayer, 10/11
Rodrigo Banuelos, 10/18
David Nualart,  10/25
Kavita Ramanan, 11/1

Workshops:

Recent Progress on Dimer Model and Statistical Mechanics, Aug. 15-16

Financial mathematics, October 13

Functional inequalities in Probability, November 2-3