Math 180: Course Summary Spring 2011, Prof. Jauregui • Probability { terminology: experiments, outcomes, events, sample space, probabilities { operations: OR, AND, NOT { mutually exclusive events { independent events, multiplication rule ∗ Case: People v. Collins { rules of probability ∗ addition rule, inclusion{exclusion ∗ subset rule, complement rule { coin flip examples (e.g., at least two heads out in 10 flips) { birthday problem { fifty-fifty fallacy { possibility of multiple lottery winners { \law of averages" fallacy { Conditional probability (definition and formula) ∗ multiplication rule for conditional probabilities { Bayes' formula ∗ disease testing, true/false negatives/positives ∗ general misunderstanding of conditional probability ∗ applications to DNA matching, determination of guilt { Prosecutor's fallacy, Sally Clark case { Defense attorney's fallacy, DNA matching, O.J. Simpson trial { Monty Hall Problem { expected value, variance, standard deviation { Bernoulli trials, special formulas for mean and variance • Statistics { bell-shaped curve/normal distribution, interpretation of µ and σ { 68 − 95 − 99:7 rule { z-scores, using the table { gathering data, computing mean and median, quartiles, IQR { Simpson's paradox (batting averages, alleged discrimination case) 1 { simple random samples ∗ sources of bias (opportunity sample, response bias, sampling bias, etc.) { yes/no surveys ∗ p andp ^, mean and standard deviation ofp ^ ∗ margin-of-error, confidence intervals • Decision theory { Drawing and solving decision trees (branches, decision nodes, chance nodes) { legal, medical, and business examples { risk aversion, certainty equivalent, Deal or No Deal • Game theory { Basics: players, games, strategies, payouts, information { dominant, dominated strategies { pure vs. mixed strategies { Nash equilibrium { 2 × 2 zero sum games ∗ maximin, minimax, checking for saddle points ∗ optimal mixed strategies ∗ value of game ∗ sports examples: baseball, soccer { m × n zero sum games ∗ elimination of dominated strategies ∗ maximin, minimax, checking for saddle points ∗ Blotto game { non-zero sum games ∗ prisoner's dilemma ∗ wage/salary negotiation ∗ Chicken ∗ Stag hunt { handout: mixed strategies in real world (price couponing, illegal parking, etc.) { moral hazard (examples, negative consequences, possible solutions) • Auctions { English auctions, Dutch auctions, first-price sealed-bid { Vickrey auctions (second-price sealed-bid), dominant strategy for bidder { all-pay auctions 2 { winner's curse • Fair division { equal division, proportional division { equal division of contested sum between two parties { Talmud and equal division of contested sum among 3 parties • Voting theory { Plurality voting (advantages/disadvantages) { Approval voting (advantages/disadvantages) { Ranked voting ∗ instant run-off (advantages/disadvantages) ∗ Borda count (advantages/disadvantages) { Condorcet criterion { Idea of clone independence of a voting system { Arrow's theorem (particularly independence on irrelevant alternatives) • Finance { Compound interest ∗ doubling time, \rule of 72" { Present value and future value ∗ single payments, annuities, perpetuities 3.