MATH 340, Study Guide for Midterm Test 1

9/30/2015

Test coverage

1. Combinatorial Analysis
   1. The Basic Principle of Counting (1.2)
   2. Permutations, Combinations, Multinomial Coefficients (1.3-5)
   3. The number of Integer Solutions of Equations (1.6: Propositions 6.1, 6.2)
2. Axioms of Probability
   1. Sample Space; Outcomes; Events. Operations on 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: Examples 5a-j,m,n; skip the rest)
   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: Examples 3acdi; Prop. 3.1, Examples 3klmn; rest can be skipped)
   3. Independent events (3.4: Examples 4abce; Prop. 4.1)
   4. Further Properties of Conditional Probability (3.5; Formula 5.1)
   5. Independent Trials (3.4: Definition of independent trials; Examples 4f,h,j; rest can be skipped)

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