Lecture Notes: Strategic Practice and Probability

Introduction to Strategic Practice

Strategic Practice involves grouping problems by themes to practice different topics.
Similar to learning chess tactics, starting with specific types (pins, forks, skewers) and later mixing them.
Emphasizes pattern recognition, which is crucial for the course.
- Importance of doing many problems for practice.

Homework should include words and explanations, not just equations.
Clarity and honesty are emphasized in reasoning.
Students should avoid sloppy arguments and should clearly justify their solutions.
- Example: Avoid jumping to solutions without explanation.
- Use sentences alongside mathematical equations.
Homework should be readable with justifications for each step.
Homework due: Beginning of class; no late submissions accepted.
- Two lowest homework scores are dropped.

Strategic Practice Problems: Serve as examples for homework expectations.
Math Review Handout: Available online, with recent updates.
Review Sessions: Fridays at 2:00 PM in Hall E; includes video recordings when possible.
Course allows pass/fail option for flexibility.

Probability is integral in many fields including physics, genetics, economics, history, and social sciences.
Example of historical application: Analysis of The Federalist Papers using probability.
Encourages exploring applications in various domains such as history, social sciences, and finance.

Historical roots of probability are in gambling and games of chance.
Important historical figures: Fermat and Pascal developed early probability concepts through letters discussing gambling.
- Their correspondence is foundational to the development of probability.

Sample Space: Set of all possible outcomes of an experiment.
Event: A subset of the sample space.
Naive Definition of Probability:
- Probability (P) of event A = (Number of favorable outcomes) / (Total possible outcomes).
- Assumes all outcomes are equally likely and finite.
- Discussed the limitations and potential misuses of this definition.

Multiplication Rule: Used to calculate the number of possible outcomes for combined experiments.
Example: Calculating the probability of a full house in poker using counting.
- Uses combinatorial logic such as "n choose k."

Discussed sampling with/without replacement and when order matters.
Filled in the sampling table for different scenarios:
- With replacement and order matters: ( n^k )
- Without replacement and order matters: ( n(n-1)...(n-k+1) )
- Without replacement and order doesn't matter: ( \binom{n}{k} )
- With replacement and order doesn't matter: More complex, ( \binom{n+k-1}{k} )
Emphasized importance of understanding these concepts for solving problems effectively.

Encouragement to start on homework and understand these fundamental concepts thoroughly.
Upcoming topics will include more on probability and counting principles.