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 ˆ, mean and standard deviation ofp ˆ ∗ 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

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