The Probabilistic Method Joshua Brody CS49/Math59 Fall 2015

Traditional Lectures • Little opportunity for feedback • Not effective way to learn

Clicker Lectures • Research: to learn, you must effectively work with a problem and construct your own understanding.

Clicker Question Why are you taking CS49/Math059? (A) like probability/randomness,want to know more. (B) Math major, course sounded interesting (C) CS major, course sounded interesting (D) fills requirement (E) other

Syllabus Highlights • Open home page • Pre-reqs: CS35, Math029 or permission from instructor • no Math029? You must pass entrance exam • Attendance Mandatory • Textbook(s) required • Alon/Spencer: the book for the Probabilistic Method • Shoup: free online, we’ll use for probability theory • Office hours, open door policy • Piazza -- all content questions go here

Labs • Lab Assignments ~every 2 weeks • Mostly problem sets, likely one programming • Final Project • Partners encouraged but not required • 2 late days per semester

What is the Probabilistic Method? Goal: show some combinatorial object has some nice property. Approach: (1) Pick a random object (2) Show that object has nice property w/prob > 0 (3) Conclude there exists an object with nice property

Example Problem Arithmetic Sequence: list of numbers (a 1 , a 2 , ..., a m ) where a i = a i-1 + k for some k examples: (1, 5, 9, 13), (201, 402, 603, 804, 1005) Problem: show how to color numbers {1, 2, 3, 4, ..., 2015} using 4 colors so that no arithmetic sequence is monochromatic Solution: color each number 1, 2, ..., 2015 randomly.

Clicker Question What is the probability that one sequence a 1 , a 2 , ..., a 11 is monochromatic? (A) 4 -11 (B) 2 -11 (C) 2 -20 (D) depends on the sequence

What is the Probabilistic Method? Goal: show some combinatorial object has some nice property. Approach: (1) Pick a random object (2) Show that object has nice property w/prob > 0 (3) Conclude there exists an object with nice property What do we learn from Probabilistic Method? • a nice solution must exist • good understanding of why nice solution exists? What don’t we learn? • how to find nice object

The Probabilistic Method • Invented by Paul Erd ő s [1913-1996] • 1500+ papers • 500+ collaborators • only interested in mathematics • itinerant 2nd half of life • Erd ő s number: publication distance from Erd ő s

Course Overview 1. Why take CS49/Math059? 2. What is this course about? 3. Course Goals 4. Resources 5. Where we’re going

The Probabilistic Method

The Probabilistic Method Joshua Brody CS49/Math59 Fall 2015 - PowerPoint PPT Presentation

The Probabilistic Method Joshua Brody CS49/Math59 Fall 2015 Traditional Lectures Little opportunity for feedback Not effective way to learn Clicker Lectures Research: to learn, you must effectively work with a problem and

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