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STAT151 (146)


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University of Alberta
Paul Cartledge

5.1 Introducing Probability Def’n: An experiment is a process that, when performed, results in one and only one of many observations. These observations are called the outcomesof the experiment. Probability is a numerical measure of likelihood that a specific outcome occurs. 3 Conceptual Approaches to Probability: 1) Classical probability - equally likely outcomes exist when two or more outcomes have the same probability of occurrence - classical probability rule: P(A) = (# of outcomes favourable to A) / (total # of outcomes for experiment) 2) Relative frequency concept of probability - experiment repeated n times to simulate probability - relative frequencies are NOT probabilities, they only approximate them. - Law of Large Numbers: If an experiment is repeated again and again, the prob. of an event obtained from the relative frequency approaches the actual or theoretical prob. 3) Subjective probability - subjective probability is the degree of belief that an outcome will occur, based on the available information 5.2/5.3 Calculating Probability Def’n: A sample space (a.k.a. S) is the set of all outcomes of an experiment. evennt (a.k.a. A) is a subset of elementary outcomesA ⊂ S. Æ P(A) = probability that A occurs • A union of 2 events is denoted by A or B (or AU B). • An intersection of 2 events is denoted by A and B (or AI B). C • A complement of an event is denoted by A . A Venn diagram is a picture that depicts S (events above drawn in class). Experiment Outcomes Samapcle Toss a coin Head, Tail S = { H, T } Toss 2-headed coin Head S = { H } Toss a $5 bill Get it back, Lose money S = { Lucky, Not Smart } Pick a suit Spades, Clubs, Diamonds, Hearts S = { Sp, C, D, H } (Associated Venn diagrams drawn in class) Properties for calculating probabilities: 1. 0 ≤ P(A) ≤ 1 2. P(A) is the sum of probabilities of all elementary outcomes comprising A. 3. P(S) = 1 Basic Rules for Finding Probability of a Pair of Events: Table 5X0 – 2-way table of responses Hockey Like Indifferent Dislike Hockey Total (A) (B) (C) Male (M) Female (F) Total Def’n: Marginal probability is the probability of a single event without consideration of any other event. Ex5.1) P(M) = P(F) = P(A) = P(B) = P(C) = Conditionalbility is the probability that an event will occur given that another event has already occurred. If A and B are 2 events, then the conditional probability of A given B is written as P(A | B). P(A| B) = P(AI B) and P(B | A) =P(BI A) P(B) P(A) such that P(A) ≠ 0 and P(B) ≠ 0. Ex5.2) P(like hockey | male) = P(A and M) / P(M) = P(female | indifferent) = P(F and B) / P(B) = Disjoint events are events that cannot occur together. caddex5.5.4) R = get red suit = {diamond, heart} E = even = {2, 4, 6} B = get black suit = {spade, club} O = odd = {1, 3, 5} F = get face card = {jack, queen, king} Pr = prime = {2, 3, 5} Which pairs are disjoint? Two events are independent if the occurrence of one does not affect the probability of the occurrence
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