Chapter 1. Probability

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1 Chapter. Probability. Set defiitios 2. Set operatios 3. Probability itroduced through sets ad relative frequecy 4. Joit ad coditioal probability 5. Idepedet evets 6. Combied experimets 7. Beroulli trials 8. Summary. Set Defiitios Set A elemet a A a A Tabular method Rule method A = {2, 4,6,8,0, } A = { a a is eve} = the set of atural umbers Z = the set of itegers Q = the set of ratioal umbers R = the set of real umbers 2

2 . Set Defiitios Def: Set A is fiite, if A has a fiite umber of elemets. ot fiite = ifiite Def: Set A is coutable, if there is a oto fuctio f : A. ot coutable = ucoutable fiite coutable A = {2,4,6} is coutable., Z, & Q are coutable. R Q( = the set of irratioal umbers) is ucoutable. R is ucoutable. 3. Set Defiitios Def: Set A is a subset of set B, if a A a B. A B Def: (proper subset) φ = ull set A B = φ A & B are disjoit. (mutually exclusive) 4 2

3 . Set Defiitios Ex.-: A = {,3,5,7}, B = {,2,3, }, C = {0.5 < c 8.5} D = {0.0}, E = {2, 4,6,8,0,2,4}, F = { 5.0 < f 2.0} tabularly specified -- rule-specified -- fiite -- ifiite -- coutable -- ucoutable -- A B A E = φ D φ Uiversal set = S rollig a die S = {, 2,3, 4,5, 6} toss two cois S = { HH, HT, TH, TT} 5.2 Set Operatios Ve Diagram 6 3

4 .2 Set Operatios A B & B A A = B A B = { x : x A & x B} = A B B = S B complemet S = φ A = A 2 2 = A A A = A A A A = A = A B = B A A B = B A ( A B) C = A ( B C) = A B C commutative associative 7.2 Set Operatios ( A B) C = A ( B C) = A B C A ( B C) = ( A B) ( A C) A ( B C) = ( A B) ( A C) distributive ( A B) = A B ( A B) = B A De Morga's Laws Duality φ S 8 4

5 .3 Probability Itroduced Through Sets ad Relative Frequecy Defiitio of Probability - set theory ad axioms - relative frequecy Experimet - sample space S - evet - probability P ( σ -algebra Ω) ( S, Ω, P) -- probability space Rollig a die. S = {,2,3,4,5,6} A = {,3,5} B = {2,4,6} C = {,2,3,4} P( A) = P( B) = P( C ) = Probability Itroduced Through Sets ad Relative Frequecy Sample space S set of all possible outcomes i the experimet Rollig a die S = {,2,3,4,5,6} Toss a coi Toss 2 cois S = { H, T} S = { HH, HT, TH, TT} Choose a umber i [0,] S = { s : 0 s } Choose a umber i S = = {,2,3, } discrete & cotiuous fiite & ifiite coutable & ucoutable 0 5

6 Evet.3 Probability Itroduced Through Sets ad Relative Frequecy = a subset of S σ -algebra Ω = the set of all evets Toss a coi S = { H, T} Ω = { φ,{ H},{ T}, S} Toss 2 cois S = { HH, HT, TH, TT} Ω = { φ,{ HH},{ HT},,{ HH, HT, TH}, S} Choose a umber i [0,] S = { s : 0 s } Choose a umber i S = = {,2,3, } A = { a : 0 < a < 0.5} A = {,3,5, }.3 Probability Itroduced Through Sets ad Relative Frequecy Probability P = a fuctio from Ω to [0,] Toss a coi S = { H, T} P( φ ) = 0, P({ H}) = 0.6, P({ T}) = 0.4, P( S) = Probability Axioms. P( A) 0 2. P( S ) = 3. A B = φ P( A B) = P( A) + P( B) 3'. mutually exclusive { A : =,2, } P( A ) = P( A ) = = 2 6

7 .3 Probability Itroduced Through Sets ad Relative Frequecy S = = {,2,3, } Properties: P({ }) = P({ }) = S = { s : 0 s } 0 a < b P({ x : a < x < b}) = b a P({0.5}) = 0, P((0.3,0.7]) = 0.4 P( A) = P( A) P( A B) = P( A) + P( B) P( A B) Probability Itroduced Through Sets ad Relative Frequecy Ex.3-2: Probability as a relative frequecy coi toss P({ H}) = lim H Ex.3-3: 4 7

8 .4 Joit ad Coditioal Probability Joit probability P( A B) Coditioal probability of evet A, give evet P( B) 0 P( A B) P( A B) = P( B) Properties:. P( A B) 0 2. P( S B ) = 3. A C = φ P( A C B) = P( A B) + P( C B) B P( B) 4. A B = φ P( A B) = Joit ad Coditioal Probability Ex.4-: A: draw a 47 Ω resistor B : draw a 5% tolerace resistor C : draw a 00 Ω resistor P( A) =, P( B) =, P( C) = P( A B) = 00 P( A C) = 0 P( A B) P( A B) = = = P( B) P( A C) 0 P( A C) = = = 0 P( C)

9 .4 Joit ad Coditioal Probability Total probability A = A S = A ( B ) = ( A B ) mutually exclusive = = P( A) = P[ ( A B )] = P( A B ) = = 7 B B 2.4 Joit ad Coditioal Probability Bayes' Theorem P( B ) ( ) ( ) A P A B P B P( B A) = = P( A) P( A) Ex.4-2: = before the chael =0 before the chael A = after the chael A 2 =0 after the chael P( A B ) P( B ) P( B A) = P( A B ) P( B ) + + P( A B ) P( B ) 8 9

10 .4 Joit ad Coditioal Probability P( B A ) =? P( A ) = P( A B ) P( B ) + P( A B ) P( B ) = = P( B A ) P( B A ) P( A B ) P( B ) P( A ) P( A ) = = = = P( B A ) =? = P( B A ) 2 P( B A 2) =? P( B2 A 2) =? 9.5 Idepedet Evets Def: Two evets A & B are (statistically) idepedet if P( A B) = P( A) P( B), P( A) 0, P( B) 0 P( B A) P( B) P( A) A & B idepedet P( B A) = = = P( B) P( A) P( A) B = B S = B ( A A) = ( B A) ( B A) P( B) = P( B A) + P( B A) A & B idepedet P( B A) = P( B) P( B A) = P( B) P( B) P( A) = P( B)[ P( A)] = P( B) P( A) A & B idepedet 20 0

11 .5 Idepedet Evets Def: 3 evets A, A2, & A3 idepedet P( Ai ) 0 P( A A2 ) = P( A ) P( A2 ) P A2 A3 = P A2 P A3 ( ) ( ) ( ) P( A A3 ) = P( A ) P( A3 ) P( A A2 A3 ) = P( A ) P( A2 ) P( A3 ) Ex: S = {,2,3,4} A = {,2}, A = {2,3}, A = {,3} 2 3 P( A A ) = P( A ) P( A ), i j i j i j A, A, & A pairwise idepedet 2 3 P( A A2 A3 ) P( A ) P( A2 ) P( A3 ) A, A2, & A3 OT idepedet Fact: A, A, & A idepedet A & ( A A ) idepedet Fact: A, A, & A idepedet A & ( A A ) idepedet Combied Experimets 2 idepedet experimets ( S, Ω, P ) & ( S2, Ω2, P2 ) Ca defie a combied probability space ( S, Ω, P) S = S S = {( s, s ) : s S, s S } Ex.6-: Ex.6-2: Ω = Ω Ω = { A A : A Ω, A Ω } Ex.6-3: P( A A ) = P ( A ) P ( A ), A Ω, A Ω P( A S ) = P ( A ) P ( S ) = P ( A )

12 .6 Combied Experimets 3 idepedet experimets ( S, Ω, P), i =,2,3 i i i Ca defie a combied probability space ( S, Ω, P) S = S S2 S3 Ω = Ω Ω 2 Ω3 P A A A P A P A P A A ( 2 3) = ( ) 2( 2) 3( 3), i Ωi Permutatio Combiatio P = ( ) ( r + ) = r! r = r! ( r)! biomial expasio! ( r)! ( x + y) = x y r= 0 r r r 23.7 Beroulli Trials Basic experimet - 2 possible outcomes ( A or A) P( A) = p P( A) = p Beroulli Trials - repeat the basic experimet times (Assume that elemetary evets are idepedet for every trial.) k P({ A occurs exactly k times}) = p ( p) k Ex.7-: P( A ) = 0.4 = 3 P (2 hits) = 0.4 ( 0.4) = k 24 2

13 .7 Beroulli Trials P (3 hits) = 0.4 ( 0.4) = P({carrier suk}) = P(2 hits) + P(3 hits) = Ex.7-3: P( A ) = 0.4 = 20 P (50 hits) = 0.4 ( 0.4) =? 20! =? large De Moivre-Laplace approximatio Poisso approximatio 25 3

What is Probability?

Quatificatio of ucertaity. What is Probability? Mathematical model for thigs that occur radomly. Radom ot haphazard, do t kow what will happe o ay oe experimet, but has a log ru order. The cocept of probability