MATH 340, Study Guide for Midterm Test 1

3/2/2014

Test coverage

1. Combinatorial Analysis
   1. Introduction: the Basic Notions of Probability; Birthday Problem (2.5, [5i])
   2. The Basic Principle of Counting (1.2)
   3. Permutations, Combinations, Multinomial Coefficients (1.3-5)
   4. The number of Integer Solutions of Equations (1.6)
2. Axioms of Probability
   1. Sample Space and Events (2.2)
   2. Axioms of Probability (2.3)
   3. Properties of Probability (2.4)
   4. "Classical Definition of Probability"; Sample Spaces with Equally Likely Outcomes 2.5
   5. Probability as a Measure of Belief (2.7: read this section)
   6. Skip Section 2.6
3. Conditional Probability and Independence
   1. Conditional probability: definition and basic properties (3.1-2)
   2. Formula of compound probabilities. Bayes' formula (3.3)
   3. Independent events (3.4)
   4. Further Properties of Conditional Probability (3.5)
   5. Independent Trials (3.4)

Key concepts

• Basic principle of counting, permutations, combinations, binomial and multinomial coefficients
• Sample space, elementary outcomes, events
• Union, intersection, complement; mutually exclusive events
• Axioms of Probability
• "Classical definition of Probability" (equally likely outcomes)
• Conditional Probability
• Formula of Compound Probabilities
• Bayes' formula
• Independent events. Multiplication rule
• Independence for more than two events. Independent trials

Important Techniques (highlights from the problems)

• Combinatorial analysis (permutations, combinations, basic principle of counting)
• Manipulating intersections and unions, proving basic relations between operations on events (DeMorgan laws, etc.)
• Matching Problem: Counting intersections and unions
• Conditional probability: formulating hypotheses, using compound probabilities and Bayes' rule
• Selection with and without replacement. Urn models
• Card play problems: A deck of 52 cards has cards of 13 denominations ("2"-"10","J","Q","K","A"), in 4 different suits ("spades", "clubs", "diamonds", "hearts"). A poker hand is a selection of 5 cards without replacement. A bridge deal is distributing the whole deck to 4 different players ("N", "S", "W", "E"), with each player receiving 13 cards.
• Independent Trials: Probability of "A before B". Using geometric series
• Independent Trials: conditioning on the outcome of the first trial; recurrence relations for probabilities
• Gambler's Ruin Problem