Math Topics
Probability: Independence and Multiplication
Writing Topics
Precision, Rigor, and Formality
Description
Discuss the proof in the pre‐recitation assignment and how it breaks down, with emphasis on the importance of independence and how the proof obscures whether the events are independent. Practice with Additional Wrong Proofs (PDF).
Pre-Recitation Assignment
Try to identify the flaw in a wrong proof.
Instructions: When writing a proof of a correct mathematical statement, the goal is to write in a way that is correct, clear, and convincing. Read this “proof” of an incorrect/wrong mathematical statement in Wrong Proof (PDF).
As you will see, erroneous arguments in a proof are sometimes quite subtle. As you read the proof, notice which aspects of the writing aid clarity (helping you to follow the author’s thinking) and which aspects make the logic seem convincing (thus causing the author to think his/her flawed proof is correct). Also think about how the author could have recognized that the proof is flawed.
As you respond to the questions below, you may use LaTeX by using “Insert equation” and clicking “Directly edit LaTeX.”
Question 1: Show that the theorem is wrong by listing the 16 possible sequences of coin flips for \(n=4\) and by using your list to calculate \(\mathbb{P}(A\wedge B)\) and \(\frac{1}{2}-\frac{f_n}{2^n}\).
Question 2: Try to find the flaw(s) in the proof. Use your list from Question 1 to try to identify which statements are false, and try to identify the logical flaw(s). We’ll go over this in recitation, and the flaw(s) are rather subtle, so you don’t need to spend too much time on it; but at least ensure you understand the proof strategies.
Post-Recitation Assignment
Write a rigorous probability proof: Problem 1(a) of Homework 2 (PDF).
