Chapter 2 - Chapter 2 Probability Study of randomness and...

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Chapter 2 Probability Study of randomness and uncertainty 2.1 sample space and event (Random) Experiment – any action or process whose outcome is subject to uncertainty eg. : tossing a coin Def) sample space ( S ) – set of all possible outcomes Toss a coin : S ={H, T} Examine function of a component : S ={S, F} Examine two fuses : S ={NN, ND, DN, DD} Ex 2.3) two gas stations with six gas pumps each. Check how many pumps are used at the two stations. - ordered pairs - S ={(0,0), (0,1), …, (6,6)} Ex 2.4) battery is determined as ‘F’ or ‘S’ according to the voltage. An experiment is to test the batteries until a first success is observed. - S ={S, FS, FFS, FFFS, …} 1
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Def) event – a collection of outcomes in the sample space Simple event, compound event Ex 2.6) – continuation of Ex 2.3 - 49 simple events - A={no of pumps in use is the same for both stations} ={(0,0),(1,1),…,(6,6)} - B={total no of pumps in use is 4} ={(0,4),(1,3),(2,2),(3,1),(4,0)} - C={at most one pump is in use} ={(0,0),(0,1),(1,0),(1,1)} Ex 2.7) – continuation of Ex 2.4 - infinite no of simple events - A={at most three batteries are examined} ={S,FS,FFS} - B={even no of batteries are examined} ={FS, FFFS, FFFFFS, …} Def) A, B : events Union of A & B : U A B - events that are either in A or B Intersection of A & B : I A B - events that are in both A and B Complement of A : A - events that are not in A 2
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Ex 2.8) – continuation of Ex 2.3 in which the first station is observed. A={0,1,2,3,4} B={3,4,5,6} C={1,3,5} --------------------------- A U B={0,1,2,3,4,5,6}= S A U C={0,1,2,3,4,5} A I B={3,4} A I C={1,3} A ' ={5,6} {A U C} ' ={6} Ex 2.9) – continuation of Ex 2.4 A={S, FS, FFS} B={S, FFS, FFFFS} C={FS, FFFS, FFFFFS, …} -------------------------- A U B={S, FS, FFS, FFFFS} A I B={S, FFS} A ' ={FFFS, FFFFS, FFFFFS, …} C ' ={S, FFS, FFFFS, …} B I C=? – :null set Def) disjoint (mutually exclusive) - when two events have no outcome in common 3
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2.2 Axioms, Interpretations and properties of probability Probability under a sample space – to assign a number P(A) to an event A Axioms of probability 1. P(A) 0 2. P(S) = 1 3. If L 1 2 , , A A mutually exclusive events, then 1 2 1 ( ) ( ) i i P A A P A = = L U U Ex 2.11) - toss a coin S ={H, T} P(H)=0.5. P(T)=0.5 P( S )=P({H,T})=1 P(H U T)=P(H)+P(T)=1 Ex 2.12) – continuation of Ex 2.4 E i ={the first satisfactory battery appears at i-th time} P(any battery is satisfactory)=0.99 P(E i )=(0.01) 1 i - (0.99) 1=P( S )=0.99+(0.01)(0.99)+(0.01) 2 (0.99)+…=1 Interpreting probability : 4
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What does it mean that we have 70% chance of rain tomorrow ? 1. we will have rain tomorrow 2. we will not have rain tomorrow 3. we will have rain for about 17 hours and no rain for about 7 hours tomorrow 4. we will have rain tomorrow as likely as we pick a red ball from a box which contains 7 red and 3 black balls 5. none of above Consider tossing a coin n times repeatedly, P(H)=0.5 ?
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This note was uploaded on 01/06/2011 for the course STAT 511 taught by Professor Bud during the Fall '08 term at Purdue University-West Lafayette.

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Chapter 2 - Chapter 2 Probability Study of randomness and...

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