Lecture on Strategic Practice in Problem-Solving

Overview

• Strategic Practice: Problems grouped by theme to practice specific topics.
• Pattern Recognition: Important for problem solving, improves with practice.

Homework Guidelines

• Expectations: Write clear, detailed solutions with reasoning and justifications.
• Incorrect Examples: Avoid sloppy, unclear solutions.
• Preferred examples: Use words and sentences to explain steps.
• Clarity and Honesty: Important to be honest and clear in your work.
• No Late Homework: Solutions are posted soon after submission. Two lowest scores are dropped.

Review Sessions

• Math Review Handout: Updated version available, includes relevant materials.
• Review Sessions: Fridays at 2:00 PM in Hall E, optional but useful for those rusty in math.
• Video Recording: Reviews are videotaped and posted online.

Possible Applications of Probability

• Physics: Quantum mechanics uses probability extensively.
• Genetics: Essential for understanding genetic variations.
• Economics and Game Theory: Probability is crucial in these fields.
• History: Example of Mosteller-Wallace study on The Federalist Papers authorship.
• Social Sciences and Government: Applications in political science and history.
• Finance: Recommended course STAT 123 for those interested.
• Gambling: Historical roots of probability in gambling games.
• Life: Probability and statistics are essential for dealing with uncertainty.

Naive Definition of Probability

• Sample Space: Set of all possible outcomes of an experiment.
• Event: A subset of the sample space.
• Naive Definition: Probability of an event = (number of favorable outcomes) / (total number of possible outcomes).
• Assumptions: Equally likely outcomes and a finite sample space.
• Limitations: Cannot be applied if outcomes are not equally likely or infinite.

Counting Principles

• Multiplication Rule: For combined experiments, the total number of outcomes = product of the number of outcomes for each experiment.

Ice Cream Example

• Choices: Cone type (cake, waffle) and flavor (chocolate, vanilla, strawberry).
• Tree Diagram: Visual representation for counting possibilities.
• Results: Total outcomes = 2 (cone choices) × 3 (flavor choices) = 6 outcomes.

Binomial Coefficient (n choose k)

• Definition: Number of ways to choose k items out of n = n! / [(n-k)!k!].
• Application: Used to determine probability in card games like poker.

Poker Example: Full House

• Definition: Hand with three of one rank and two of another.
• Calculation: Use multiplication rule to count number of possible full house hands.

Sampling Methods

• With Replacement and Order Matters: Total outcomes = n^k.
• Without Replacement and Order Doesn't Matter: Total outcomes = n choose k.
• Without Replacement and Order Matters: Total outcomes = n × (n-1) × ... × (n-k+1).
• With Replacement and Order Doesn't Matter: Total outcomes = (n+k-1) choose k (will be proven).

Summary

• Homework: Start early, most can be completed with material covered so far.
• Further Topics: More on sampling and probability next time.