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Chapter 1 Introduction • Use of probability in daily life A. B. C. D. Lotto Batting average in baseball Election Poll (statistics) Weather Forecast tch-prob 1 Why is Probability important • • • Many things change all the time – Signal strength (voltage/current) – Life of light bulb – Time of Rain – Traffic arrival Probability can be used to establish mathematical model so that these phenomena can be analyzed in a quantitative way. Applications: – – Signal variations in communication system (due to error occurrence) Traffic congestion problem (when input will occur) tch-prob 2 Various definitions of probability A. B. C. D. Relative-frequency definition Classical definition Probability as a measure of belief Axiomatic definition tch-prob 3 A. Relative-frequency definition The experiment under consideration is repeated n times. If the event A occurs na times, then its probability P(A) is defined as the na limit of the relative frequency n of the occurrence of A. n P( A) lim na n Ex. Toss a coin Comment. 1. based on experimental ground. natural na 2. n is finite, limit of n cannot be obtained This definition is a hypothesis about the existence of the limit. tch-prob 4 B. Classical definition The probability P(A) of an event A is found a priori without actual experimentation. If N is the total number of possible outcomes of the experiment, and in N a of these outcomes the event A occurs, then P( A) Na N Ex1. Roll a die Pr [even] = 36 Ex2. Roll a pair of dies p = Pr [ Sum of two outcomes = 7] = ? 1. There are 11 possible sums:2, 3, ….,12 p= 1 11 tch-prob 5 2. Count all possible pairs of numbers, not distinguishing between the first and second die. (1,1) (1,2) (1,3)(2,2) (1,4)(2,3) (1,5)(2,4)(3,3) total 21 (1,6)(2,5)(3,4) ( 2,6 ) ( 3,5 ) ( 4,4 ) 3 p = 21 ( 3,6 ) ( 4,5 ) (4,6 ) ( 5,5 ) ( 5,6 ) Note: possible outcomes are not equally likely. (1,2)is more likely than(1,1) ( 6,6) tch-prob 6 3. Count all pairs; distinguish between 1st and 2nd die. There are 36 possibilities, of which 6 have sum = 7 (3,4)(4,3)(5,2)(2,5)(1,6)(6,1) p= 6 36 Improved version of the classical def: N P( A) a , provided all N outcomes are equally likely . N tch-prob 7 Criticism of the classical definition. 1. The definition is circular. What is “equally likely”. 2. can be used only for a limited class of problems Ex. unfair coin, Pr [5] = 1 , … 4 3. make implicit use of the relative-freq. interpretation of probability. Pr [5] 1 6 4. number of outcomes may be infinite. tch-prob 8 Ex. (Bertrand Paradox) In a circle C of radius r we draw at “random” a cord AB. What is the prob. that the length l of this cord is greater than the length of the side of an inscribed equilateral triangle? Solution. a. The center M of the cord AB is a “random” point. If it lies inside the circle C1 of radius r/2 then C l 3r M A P r2 B r/2 4 1 r2 4 C1 r tch-prob 9 b. Fixed point A If B lies on the arc DBE, Then D l 3r B 2 r P 3 1 2 r 3 A tch-prob E 10 c. Assume AB is perpendicular to the diameter FK. l 3r if the center M of AB lies between G and H. F r 2 G M A P r 1 2r 2 B H K tch-prob 11 • The above three solutions correspond to three different experiments. • The example shows the difficulties associated with the classical definition. • It is meaningless to talk about the prob. of an event; the underlying experiment must also be clearly specified. tch-prob 12 C. Probability as a measure of Belief that something might or might not be true. Ex.” It is probable that X is guilty ”. • It is a statement about a single event, and not about averages of mass phenomena. • This concept of probability is a form of inductive reasoning. Induction: inference of a generalized conclusion from particular instances deduction: the deriving of a conclusion by reasoning inference in which the conclusion about particulars follows necessarily from general or universal premises. tch-prob 13 D. Axiomatic Definition The probability of an event A is a number P(A) assigned to this event. This number obeys the following three postulates but is otherwise unspecified: I. P(A) is positive: P( A) 0 II. The probability of the certain event S equals 1 : P(S)=1 III. If A and B are mutually exclusive, then P A B P A P B The resulting theory is based on these postulates and on nothing else. tch-prob 14 • • Axiom: 1. a fundamental principle widely accepted on its intrinsic merit 2. a statement accepted as true as the basis for argument or inference Postulate: 1. a hypothesis advanced as an essential presupposition, condition, or premise of a train of reasoning. tch-prob 15