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Lecture 1: Introduction
Lecture 2: Introduction (cont.)
Lecture 3: Linearity of Expectations
Lecture 4: Linearity of Expectations (cont.)
Lecture 5: Alterations
Problem set 1 due
Lecture 6: Second Moment
Lecture 7: Second Moment (cont.)
Lecture 8: Chernoff Bound
Lecture 9: Chernoff Bound (cont.)
Problem set 2 due
Lecture 10: Lovász Local Lemma
Lecture 11: Lovász Local Lemma (cont.)
Lecture 12: Special lecture by Joel Spencer
Problem set 3 due
Lecture 13: Lovász Local Lemma (cont.)
Lecture 14: Lovász Local Lemma (cont.)
Lecture 15: Correlation Inequalities
Lecture 16: Janson Inequalities
Problem set 4 due
Lecture 17: Janson Inequalities (cont.)
Lecture 18: Concentration of Measure
Lecture 19: Concentration of Measure (cont.)
Lecture 20: Concentration of Measure (cont.)
Problem set 5 due
Lecture 21: Concentration of Measure (cont.)
Lecture 22: Concentration of Measure (cont.)
Lecture 23: Concentration of Measure (cont.)
Lecture 24: Entropy
Lecture 25: Entropy (cont.)
Problem set 6 due
Lecture 26: Entropy (cont.)
Lecture 27: Containers
Instructor Interview
Below, Professor Yufei Zhao describes various aspects of how he taught 18.226 Probabilistic Methods in Combinatorics.
OCW: You structured problem sets a little differently in this course, providing students with a single file with many problems but only requiring a subset of these problems to be turned in for assessment. Tell us about your decision to structure problem sets this way.
Yufei Zhao: I wanted to give the students lots of opportunities to practice the techniques taught in lectures. The single-file format made it easier to add new problems to the list as the semester progressed, and in a way that was synchronized with the pace of the lectures. I only required the students to turn in a subset of the problems in order to ease the workload burden.
OCW: How did you use the lecture time in class to ensure maximum benefit to your students?
Yufei Zhao: The goal of the course was to introduce students to the probabilistic method, which has many applications that are quite nice and short. The students appreciated seeing a lot of examples in class. Longer proofs tend to be less well suited to the lecture format, as they’re harder to follow. I’ve found that the lectures that work best are ones filled with small and neat examples and applications illustrating the method being discussed.
Curriculum Information
Prerequisites
- 18.211 Combinatorial Analysis
- 18.600 Probability and Random Variables
- 18.100A Real Analysis, 18.100B Real Analysis, 18.100P Real Analysis, or 18.100Q Real Analysis
or permission of instructor
Requirements Satisfied
18.226 can be applied toward a doctorate degree in Pure or Applied Mathematics, but is not required.
Offered
Every other fall semester.
Assessment
Grade Breakdown
Grading for 18.226 was primarily based on homework grades (no exams). A modifier up to 5 percentage points may be applied (in either direction) in calculating the final grade, based on factors such as participation.
Student Information
Enrollment
41 students
Breakdown by Year
The class included a mixture of graduate students and undergraduates.
Breakdown by Major
Students in the course came primarily from mathematics and computer science.
Typical Student Background
Most students in the course had strong mathematical problem solving backgrounds; many of the undergraduates had experience participating in math competitions.
How Student Time Was Spent
During an average week, students were expected to spend 12 hours on the course, roughly divided as follows:
Lecture
Met 2 times per week for 1.5 hours per session; 25 sessions total.
Out of Class
Outside of class, students spent most of their time solving problems from the six assigned problem sets.
The problem sets (PDF) are a list of problems for practice. A subset of these problems will be designated as to-be-turned-in. Only the designated problems are required to be submitted. Bonus problems, marked by ★, are more challenging. A grade of A- may be attained by only solving the non-starred problems. To attain a grade of A or A+, you should solve a substantial number of starred problems.
PROBLEM SET | PROBLEMS |
---|---|
Problem set 1 | A3, A8, A9, A10★, A11 (a, b★), A12★, A14, A15, A16★, and A17★ |
Problem set 2 | B4, B5, B7★, B8★, B9★, C1, C3, C4, C6★, and C7 |
Problem set 3 | C8, C10, C11★, C12★, D2–D4, D5 (a★, b★), E1, and E3★ |
Problem set 4 | E4, E6, E7, E8★, E9★, E10(a, b★), E11(d★), E12, and E13★ |
Problem set 5 | F3 (a, b), F6, F7★, F8★, G1, G3 (a), G4★, G6★, H1, H4★, and H5★ |
Problem set 6 | H7★, H9★, H10, H11, H13, H14★, I3★, I4, I5, I6★, I7★, and I10★ |
Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Prerequisites
- 18.211 Combinatorial Analysis
- 18.600 Probability and Random Variables
- 18.100A Real Analysis, 18.100B Real Analysis, 18.100P Real Analysis, or 18.100Q Real Analysis
or permission of instructor
Course Description
This course is a graduate-level introduction to the probabilistic methods, a fundamental and powerful technique in combinatorics and theoretical computer science. The essence of the approach is to show that some combinatorial object exists and prove that a certain random construction works with positive probability. The course focuses on methodology as well as combinatorial applications.
Topics
- Linearity of expectations
- Alterations
- Second moment
- Chernoff bound
- Lovász local lemma
- Correlation inequalities
- Janson inequalities
- Concentration of measure
- Entropy
- Containers
Textbook
Alon, Noga and Joel H. Spencer. The Probabilistic Method. Wiley, 2016. ISBN: 9781119061953. (The fourth edition is the latest, but earlier editions suffice.)
Grading
Grading is primarily based on homework grades (no exams). A modifier up to 5 percentage points may be applied (in either direction) in calculating the final grade, based on factors such as participation.
Final letter grade cutoffs: Only non-starred problems are considered for the calculations of letter grades other than A and A+.
- A− : ≥ 85%
- B− : ≥ 70%
- C− : ≥ 50%
Grades of A and A+ are awarded at instructor’s discretion based on overall performance. Solving a significant number of starred problems is a requirement for grades of A and A+.
Note that for MIT students, ± grade modifiers do not count towards the GPA and do not appear on the external transcript.