probnotes2 - TABLE I . Tmnslafiarls between mnls and sels....

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Unformatted text preview: TABLE I . Tmnslafiarls between mnls and sels. To imerpref the Venn diagrams in lerrns of events, imagine flunk a poinl is picked al random From Ihe Squaw. Each paint in the squwe then represenrs an outcome, and each region of Ihe diogmm represema Ihe evem H10: Ihe paint is picked ham 1ha! region. Event language Set language Set notation Venn diagram outcome space universal set [I - event 'subsel of H A. B. C. etc. IE impossible event empty set E I no: A. opposite of A complement of A A‘ either A or B or both union of A and B A U B A.3 ham A and B intersection of A and 3 AB, A n .3 w A and B are mulually cxclusivc A and H are disioim AB = G b b if A then B A Is a subset of 8' A g B B W—fi— 72 Chapter I. Introduction Introduction: Summary Outcome space: A set of all possible outcomes of a situation or experiment. such that one and only one outcome must (recur. Events: Represenled as subsets ofan outcome space. A and 3. AB. A n B. intersection event that both A and B occur. A or B, A U B, union.- event that either A or B (or both) occur. AB = @, disjoint: mutually exclusive: no overlap, no intersection. not A, A‘, complement: opposite of A: event that occurs if A does not. A C B, inclusion A is a pan of B. A implies B. ifA occurs then so does 3. 11, whole sell, outcome space: certain event. all possibilities. sure to happen. fl. empty set. impossible event: no way.r to happen. partition of A: disjoint sets A1, . .. ,A,‘ with union A. Rules of Probability and Proportion I Non-negative: PU” 2 0 . Addition: PM) = L1 PfA;)ifA1,...,A,. is a partition of A I Total of 1: 13(9) = 1. I Between 0 and 1: [I S PD!) 3 1 I Empty Set: PM) = D I Complements: PM‘) = 1 v PUD I Difference: P[BA”} = PLB) — PU!) if A C B I Inclusion—Exclusion: Phi or B) = Pill) + P[B) — P(AB). Relative frequency: Proportion of times something happens: W Interpretations of Probability I long-run relative frequency (statistical average): FAA} 96 PTA) for large rt. I degree of belief (probabilistic opinion) Probability distribution over- it: Assignment of probabilities to events represented as subsets of Q. satisfying rules of probability. A distribution over a finite set 0 can be specified with a distribution table: null III The probabilities must sum to 1 over all outcomes. Section 1.6. Sequences oi Events 73 Odds Chance odds: ratio of probabilities, e.g., the following arc equivalent: PM} = 3le; [he odds of A are 3 in 10; the odds in favor OfA are 3 to T: the odds against A are T to 3. Payofl' odds: ratio of stakes: W (what you got does not include what you bct). Flair odds mile: in a fair bet, payoff odds equal chance odds. Conditional Probability H.413) = probability of A given 8: probability of A with outcome space rcduccd to 13'. Compare with PM) = overall or unconditional probability of A. Interpretations of conditional probability: :- intuitive/subjective: chance of A if B is known to have omrrcd: a Long-run. frequency: long-run relative frequency of A's among trials that product: B. As a function of A, for fixed 3, conditional probabilities satisfy the rules of probability. eg.‘ PM‘IB) = 1 — P(A|B) Rules of Conditional Probability Division: P(A|B) = ‘23:) (note: AB = BA) For probabilities defined by counting, P[A|B) = #(AB){#{B). Similarlyr for Icngll'l. arca. or volume instead of #. Product: H.413) = PtA)P(BlA) = P(BJP(A|B] The following rules refer to a paflifion 31,. .. . B” of 0, so PiBI) +‘ '-+P(B..) = 1; for example, 3; = B. B; = ii?6 for any 3. Average rule: PM) = P(A|BI)P(B;} + + antenna“) Bayes-nae HEM} = W whore PM.) is given by the weighted average formula. Independence Two trials are independent if learning the result of one does not affect chances for the othcrI c.g.‘ two draws at random with replacement from a box of known composition. The birds are dependent if learning the result of one does affect chances for the other. e.g., two draws at random without replacement from a box of known composition. or two draws at random with replacement from a box of random composition. Independent events: A and B are such that P(AB) = P(A)P(B] ~t=> P(A[B) = PM) [learning 3 occurs does not affect chances of A} <=t PLBIA] = P[B) (learning A occurs does not affect chances of B) independence of :1 events A1,. .. ,A": P{A1A2 ' - - An) = PLAlJ ' ' ' P(An)i and the same will! any number of complements A: substituted for A‘- t?" identities), ...
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probnotes2 - TABLE I . Tmnslafiarls between mnls and sels....

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