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STAT141 (21)


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University of Alberta
Paul Cartledge

Ch. 14 – Introducing Probability Def’n: An experiment is a process that, when performed, results in one and only one of many observations (or outcomes). Probabilityis 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) Personal (or subjective) probability - personal probability is the degree of belief that an outcome will occur, based on the available information Calculating Probability Def’n: A sample space (a.k.a. S) is the set of all elementary 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 (A, B, or both happen) is denoted by A or B (or A∪ B). • An intersection of 2 events (A and B happen together) is by A and B (or A∩ B). C • A complement of an event (event does not happen) is denoted by A . A Venn diagram is a picture that depicts S (events above drawn in class). Experiment Outcomes Samacele 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, S = { AceSp, 2Sp, 3Sp,…, Hearts AceC, 2C,…, KingH } (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 Ch. 15 – Probability Rules! Basic Rules for Finding the Probability of a Pair of Events: Table 15X0 – 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. Ex15.1) P(M) = P(F) = P(A) = P(B) = P(C) = Condiptoonalbility 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). Keywords: givenf, of P(A| B) = P(A∩ B) and P(B | A) = P(B∩ A) P(B) P(A) such that P(A) ≠ 0 and P(B) ≠ 0. Ex15.2) a) If you are male in this class, what is the probability that you like hockey? P(A∩ M) P(like hockey | male) = P(A| M) = P(M) = b) What is the probability of being female in this class, given that you are indifferent to hockey? P(F ∩ B) P(female | indifferent) =(F | B) = = P(B) Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other. In other words, P(A | B) = P(A) OR P(B | A) = P(B) Ex15.3) From Table 15X0, P(F) = P(F | B) = Since probabilities are not equal, the 2 events are not independent. Ex15.4) deck of cards: P(Black) = 26/52 = ½ P(Black | Face) = 6/12 = ½ Since the probabilities ARE equal, the 2 events are independent. Disjoint (or mutually exclusive) events are events that cannot occur together. Ex15.5) deck of cards Ex15.6) a single die R = get red suit Æ diamond or heart E = even = {2, 4, 6} B = get black suit Æ spade or club O = odd = {1, 3, 5} F = get face card Æ jack, queen, or king Pr = prime = {2, 3, 5} Which pairs are disjoint? Two important observations regarding disjoint, independent & dependent events: 1. Two events are either disjoint or independent, but not both (unless one has zero prob.). 2. Dependent events may be disjoint, but disjoint events are always dependent. C Complement Rule: P(A)C+ P(A ) = 1, so C P(A) = 1 – P(A ) and P(A ) = 1 – P(A) Ex15.7) From Table 15X0, P(Fem
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