Lecture 1. (Part II) The number of ways of partitioning n distinct objects into k distinct groups containing n 1,

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1 Lecture (Part II) Materals Covered Ths Lecture: Chapter 2 ( ) The umber of ways of parttog dstct obects to dstct groups cotag, 2,, obects, respectvely, where each obect appears exactly oe group ad, s N!, 2,...,! 2! L! Example: A labor dspute has arse cocerg the dstrbuto of 20 laborers to 4 dfferet costructo obs. The frst ob (cosdered to be very udesrable) requre 6 laborers; the secod, thrd, ad fourth utlzed 4,5 ad 5 laborers, respectvely. The dspute arose over a alleged radom dstrbuto of the laborers to the obs whch placed all four members of a partcular ethc group o ob. I cosderg whether the assgmet represeted ustce, a medato pael desred the probablty of the observed evet. (). Determe the umber of sample pots the sample space S for ths expermet. (). Fd the probablty of the observed evet f t s assumed that the laborers are radomly assged to the obs. Combato: The umber of combatos of obects tae r at a tme s the umber of subsets, each of sze r, that ca be formed from the obects. Ths umber wll be deoted by r C r or. Combato Rule: The umber of uordered subsets of sze r chose (wthout replacemet) from avalable obects s! r r!( r)!

2 Codtoal Probablty ad the Idepedece of Evets. Defto: The codtoal probablty of a evet A, gve that a evet B has occurred, s equal to A A I provded > 0. Example: Suppose that a balaced de s tossed oce. Use the above defto to fd the probablty of a, gve that a odd umber was obtaed. Defto: Two evets A ad B are sad to be depedet f ay oe of the followg holds: A B A I. Otherwse, the evets are sad to be depedet. Example: Three brads of coffee, X, Y, ad Z, are to be raed accordg to taste by a udge. Defe the followg evets: A: Brad X s preferred to Y. B: Brad X s raed Best. C: Brad X s raed secod best. D: Brad X s raed thrd best. If the udge actually has o taste preferece ad radomly assgs ras to the brads, s evet A depedet of evets B, C, ad D.

3 Two Laws of Probablty (). The Multplcatve Law: The probablty of the tersecto of two evets A ad B s P ( A I B A If A ad B are depedet, the P ( A I (). Addtve Law: The probablty of the uo of two evets A ad B s A U + A I If A ad B are mutually exclusve evets, the A I 0 ad P ( A U +. (). Complemet Law: If A s a evet, the. Example: Two evets A ad B are such that 0.2, 0.3, ad A» 0.4. Fd the followg A I, A U, A I, P ( A Example: Suppose that A ad B are depedet evets such that the probablty that ether occurs s a ad the probablty of B s b, show that b a. b

4 (2). The Evet-Composto Method A summary of the steps used the evet-composto method follows: a. Defe the expermet. b. Vsualze the ature of the sample pots. Idetfy a few to clarfy your thg. c. Wrte a equato expressg the evet of terest, say, A, as a composto of two or more evets, usg uos, tersectos, ad/or complemets. d. Apply the addtve, multplcatve ad complemet laws to the compostos step c. Example: Of the voters a cty, 40% are Republcas ad 60% are Democrats. Amog the Republcas 70% are favor of a bod ssue, whereas 80% of the Democrats favor the ssue. If a voter s selected at radom the cty, what s the probablty that he or she wll favor the bod ssue. 6. The Law of Total Probablty ad Bayes Rule. () Partto of Sample Space: For some postve teger, let the sets B, B 2,, be such that. S B U B ULU B, 2. B I Φ for. B The the collecto of sets { B, B2,, s sad to be a partto of S. L } (2) The Law of Total Probablty: Assume that { B, B2,, } L s a partto of S such that ) > 0 B for all,2,,. The for ay evet A, A B ) B )

5 (3) Bayes Rule: Assume { B, B2, L, s a partto of S such that B ) > 0 for all,2,,. The } B A B ) B ) A B ) B ) Example: A electroc fuse s produced by fve producto les a maufacturg operato. The fuses are costly, are qute relable, ad are shpped to supplers 00-ut lots. Because testg s destructve, most buyers of the fuses test oly a small umber of fuses before decdg to accept or reect lots of comg fuses. All fve producto les produce fuses at the same rate ad ormally produce oly 2% defectve fuses, whch are dspersed radomly the output. Ufortuately, producto le suffered mechacal dffculty ad produced 5% defectves durg the moth of March. Ths stuato became ow to the maufacturer after the fuses bee shpped. A customer receved a lot produced March ad tested three fuses. Oe faled. What s the probablty that the lot was produced o le? What s the probablty that the lot came from oe of the four other les?