Midterm 1 - 2 Sample Spaces and Events 0 An experiment...

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Unformatted text preview: 2 Sample Spaces and Events 0 An experiment, chance experiment, or a random experiment is an activity, process, or situation Whose outcome is not known with certainty with at least 2 distinct outcomes, *7 Coin Toss * Commuting time from home. to work -- Radar signals a The set of all possible simple outcomes ie called the Sample Space denoted by S 3—: {51,527‘..gsn} Where the are simple eVents and n. is the number of events or elements of the Space I Example: Consider a System which consists of three components which either pass or fail a reliability test. \‘Vhat is the entire sample space? A I Read: Section 1.2 0 Problems in Section 1.2: 1.1, 1.3 3 Types of Probability o A priori or classical probability prior knowledge of the process or population based on rigorous inspect-ii )11 or assumptions, Suppose that we conduct. an experiment which has N different. outcomes. Suppose the experiment results in an event 1L NA < N times. Gene: ally, the probability of the EVBIIE A is the ratio [\‘rq ‘7 , p A : A m “Mr w i m ( ) 1N7 I Empii‘ical or relativefrequency probability uses observed iiifoifmaiion, information may not be complete or reproducible “Fit/:3“ 3 , - Subjective Probabiliny uses; personal information and opinion about the population; may not be basal on any information at all T-I. _‘ _ 2’" ~- r . i j, T: .4 L'x - Example: Consider a. system which consists of three components which either pass or fail a reliability test. Recall the sample space. -— \Vlmt is the probability that. one component pee-see the reliability test? I Read: Section 1.3 - Problems in Section 1.3; no problems 4 Set Theory 0 A set is a collection of objects or elements. A sample space at set: and subsets of the sample space (events) are sets Well. u The empty set is a set with no elements, Ql - Equality: Let A and B he sets if A C B and B C A; A and B are said to be equal, denoted A 2: B 3 Example: Consider system which consists of three components which either pass or fail a reliability test. Recall the sample space. Let A be the set of all outcomes where the third component passes the reliability test. Let B be the set of all outcomes where the second component passes the reliability test. Does A : B? 1 ‘I g _ . . ~=-~~ ' v ' ~ ' ~51:- / :3 . J i. ‘1 I" 1 ‘ : Iii“ ,i 1., (1 /I““:-‘ fir—i i, -JI‘J - Complements: Let S be a sample space and A be a subset of S5 A C S. The subset which consists of all of the elements of S which are not contained in A is called the compliment of A, denoted AC or A. 9 Example: Consider system which consists of three components which either pass or fail a reliability test. Recall the sample space. Let A be the set of all outcomes Where the third component passes the reliability test. What is AC? 0 Union: Let A and B he sets. The union of A and B is the set which consists of the elements in either A or B or both, denoted A U B 0 Example: Consider a system which consists of three components which either pass or fail ii reliability test. Recall the sample space» Let A be the set of all outcomes Where the third component passes the reliability test. Let B be the set of all outcomes Where the second component the reliability test \Vhet is A U B? vi ‘1 i"? ‘ im= l-I ‘.' ‘_ ?' "Hf—l :' - n. v r g _;H’-~ .‘ mm} 4 Set Theory 0 A set is a collection of objects or elements. A sample space at set: and subsets of the sample space (events) are sets Well. u The empty set is a set with no elements, Ql - Equality: Let A and B he sets if A C B and B C A; A and B are said to be equal, denoted A 2: B 3 Example: Consider system which consists of three components which either pass or fail a reliability test. Recall the sample space. Let A be the set of all outcomes where the third component passes the reliability test. Let B be the set of all outcomes where the second component passes the reliability test. Does A : B? 1 ‘I g _ . . ~=-~~ ' v ' ~ ' ~51:- / :3 . J i. ‘1 I" 1 ‘ : Iii“ ,i 1., (1 /I““:-‘ fir—i i, -JI‘J - Complements: Let S be a sample space and A be a subset of S5 A C S. The subset which consists of all of the elements of S which are not contained in A is called the compliment of A, denoted AC or A. 9 Example: Consider system which consists of three components which either pass or fail a reliability test. Recall the sample space. Let A be the set of all outcomes Where the third component passes the reliability test. What is AC? 0 Union: Let A and B he sets. The union of A and B is the set which consists of the elements in either A or B or both, denoted A U B 0 Example: Consider a system which consists of three components which either pass or fail ii reliability test. Recall the sample space» Let A be the set of all outcomes Where the third component passes the reliability test. Let B be the set of all outcomes Where the second component the reliability test \Vhet is A U B? vi ‘1 i"? ‘ im= l-I ‘.' ‘_ ?' "Hf—l :' - n. v r g _;H’-~ .‘ mm} 5 Properties of Probability a The set of all poseible outcomes is called the sample space S 0 An event is any collection or subset of the outcomes contained in S a A simple event is exactly one outcome, a. compound event consists of more than one outcome 0 Let .S' be a sample space and let A be an event. Denote the probability of A occurring as P (A) o Axiom 1: All probabilities are between 0 and l, E [0,1l for any A u Axiom 2: The total probability of the sample space is 1, P(.S') = l, the outcome will be eomewhere in the sample space a The probability of the empty is zero, P z 0 e Let A be an event and let AC be the compliment. Then P01?) :-1 ~ P H 0 Let A and B be events such that A C B. Then, P(A) S P(B), i Example: Consider a. system which consists of three components which either pass or foil 21 reliability test. Recall the sample Space, Let A be the set of all outcomes Where the third component passes the reliability test. Let B be the Set of all outcomes Where the second component paeses the reliability test. \Vhat is PUD whiff)? y P l ’i i- k l 3 2'! i: i - I 7_ .2 a " v; i .i s ’ ,i. .a g ‘ . c a . a o 3: a , . ; .1 31 . . l l,r“l“1._j P fl —.., "3’ ‘53 T—T—j/ g «i; l ‘ ' \ L e a: .I ~, . : 0 Let A and B be events. The probability of the union of A and B is given by P(ALJ B) : P(A) + 13(3) - PM n B) 0 Example: Consider a system which consists of three components which either pass or fail a reliability test. Recall the sample space. Let A be the set of all outcomes where the third component passes the reliability test, Let B be the set of all outcomes where the second component passes the reliability test. \Vhot is P(/l O B l and P(A U B)? ..- l N a { Ev - ‘ a Let A and B be mutually exclusive events such that P(A Fl B) : 0 then PM o B): PM) + 10(3) a Example: Consider a system which consists of three components Which either or fail a reliability test. Reeall the sample space. Let A be the Set of all outcomes where the third component passes the reliability test. Let B be the set of all outcomes where the second component passes the reliability test. Are A and B mutually exclusive? I Read: Section 1.6 £3 I“! a Problems in Section 1.6: Elf-2‘“ , ‘ ;-‘ " f K“) 7_ ,w“ w 7 " A 1"" L 7" L“la £15317" ri—“V A 7) ,i l 3" lE’. L“ K- 3 fl/"lb 12 H be The €V9/17L 010 Ck hazy'ic/OfiUPCHjty " g )5 QWBJ’F} C97? [’10 Mildbfig “QED -B if» we fivérfi of a 5M MAWWI e E 9% Me emf [ma m/l} 531.4)0 77x9 911M? id” :56 am H I L3) $a/vm, W): Wm. 29%) 11M) .> :2 Pawnx/ :3 FCEMJ .3 51M m 35 M a fi-fim 2: 5/1/17 HEW) L; gaming FCEPWW WK???) =9 H": '37’ PUB/13):: .) 6 Conditional Probability a Let A and B be events in 8. Suppose A depends on B, knowing B adds information about A. The conditional probability of A given B is denoted P(A]B) gig-Jpn?) : Pm—figflsg i J 0 Example: Suppose that among, people buying a digital 60% purchase an op— tional memory card, 40% purchase an optional battery, and 30% purchase both the optional memory card and battery. (a) Given a person purchases the battery, What is the probability the person also purchases the memory card? i - __ ‘ k i: ' I ' I ‘ I i r i 1‘ ), {37' (b) Given a person purchases the memory cardE What is the probability the person purchases the battery? h 1 r . L: i g :=‘ ., 7.- = J r a I: i r M u ’= 'r i ! ' j" . n x, We c a .___ Q a fix a . ; - 3‘ . ' . ‘ Hg» 3 awn ,_ . 0 Let A be a set which consists of i; disjoint subsets A1, . . . ,J—lk such that A : {A1, . . i , 11k}, A,- H A}- = 9 When i j (pairwise disjoint), and Al U . .. U Ah 2 A (the A7; form a complete partition of A). The probability of A is given by 5' m I ‘. K, PM) : Hawaii/11) + . . . HAQHA Ak) =1:- ' 2 Z PrAiiPrAiAi-i "' is; 17:] which is called the law of total probability for an event A. Here, the probability of the event A is not known directly. \Ve use the conditional probability of A given A; has occurred P(AiAz-), and the individual probability of Ah P{A.,-), - Example: The transmission of bits over a binary communication channel has the fol— lowing characteristics: (Ts and 11s have an equal chance of being sent. There is a 95% chance that a sent 0 will be received as 0. There is a 90% chance that a sent 1 Will be received as l. . ‘ ‘ \ 1:715 "' ‘ .‘ ' r ' '3' “ s; w 5 fl . If 3 c1 if? {a} Vi’lmt is the probability their 311 error will occur in thepransmiseion? w , .5: “bit __ : - - r": If F )— nfir‘ fag ‘\ “u _7 WI: lfl ltlllfiifl*. Vial ‘l"j ‘ :: «:2. w. 61:: ll .3?!“ “3" '1 --. i- “ l ’W“‘=""‘ \Jf ,‘i‘j 1L" +' V ,‘1 " x. " i \ ‘ A‘ L/av I (b) What is the probability that a 1 will be received? « . 3 ._ u “l; r» 9'": m "7 r' a» x 7, m" " " l ' ' " h l l9 f”; l “‘“~ l’ll‘i‘ ii ‘I‘f' - ‘ ' a j m g. - ..; ~_ \ _.__ g ' of] i‘ w I, n F.“ vi.“ ' f“! 39 i i “a. L “' 1‘ K ‘ “,4 0 Suppose We are in the opposite situation. The event A has been observed, but it is not lmow which of film A; events have oeeurrrxi. We can use the definition of conditional probability in combination with the law of total probability to obtain the [IUIICl‘llZlOIIal probability of A; given A _ w~vfr- P&HA' b -m' ‘- 7 Hemm _ __T“_____ ZPMi-JPMIAA ‘7 i=1 where P{A,; Fir-A) rtu obtain yir ’ 7f\.n. PAAPMi fi_i f H ‘ _ FHA/1) = ; . . i ' ZPlAilPlAlAi} r - ‘ ‘7 in: which is called Bayes’ Formula or Bayes’ Rule I Example: The transmission of bits over a. binary communication channel has the fol— lowing characteristics: 0’s and 1’s have an equal chance of being sent. There is a 95% chance that it sent 0 will be received 0. There is a 90% chance that 3 Sent 1 will be received as l. (21) Given a received 1, what is the probability of a sent 1'? 14 (1)) Given a received 9, what is the probability of a. sent 07 -._ , 0 Experiments that take place in stages (Sr succession with a finite amount. of disjoint; uumrunms with known probabilities of Dccupance can be repreaented graphically with ii Diagmm. A tree diagram is a directed non-cycfic graph. W Rout Node: graphs: begin with at least. one. of these 7 Parent Node: a sucussive node which has children 7 Child Node: a terminal node with no further splits, also cafled a leaf Branch: a path which connects nudes Resulting outcome ‘ .r" A m? P(BL’1}E (ff, x" A H [ff Ram—m may" PEER, «a f ,r f»*' 5‘ A fit: f/J/ RH» xxx " a 3 A’nB HER Egg/1f” m “a (gays A " \ ‘ Tm, HEW—H, qr aengr 15a ...
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Midterm 1 - 2 Sample Spaces and Events 0 An experiment...

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