Section 2:00 ~ 2:50 pm Thursday in Maryland 202 Sep. 29, 2005

Transcription

1 Seto 2:00 ~ 2:50 pm Thursday Marylad 202 Sep. 29, Homework assgmets set ad 2 revews: Set : P. A box otas 3 marbles, red, gree, ad blue. Cosder a expermet that ossts of takg marble from the box, the replag t the box ad drawg a seod marble from the box. Desrbe the sample spae. Repeat whe the seod marble s draw wthout frst replag the frst marble. P3. Two de are throw. Let E be the evet that the sum of the de s odd; let F be the evet that at least oe of the de lads o ; ad let G be the evet that the sum s 5. Desrbe the evets EF, E» F, FG, EF, ad EFG. P5. A system s omposed of 5 ompoets, eah of whh s ether workg or faled. Cosder a expermet that ossts of observg the status of eah ompoet, ad let the outome of the expermet be gve by the vetor ( x, x 2, x 3, x 4, x 5 ), where x s equal to f ompoet I s workg ad s equal to 0 f ompoet I s faled. (a) How may outomes are the sample spae of ths expermet? (b) Suppose that the system wll work f ompoets ad 2 are both workg, or f ompoets 3 ad 4 are both workg, or f ompoets, 3, ad 5 are all workg. Let W be the evet that system wll work. Spefy all the outomes W. () Let A be the evet that ompoets 4 ad 5 are both faled. How may outomes are otaed the evet A? (d) Wrte out all the outomes the evet AW. P6. A hosptal admstrator odes omg patets sufferg gushot wouds aordg to whether they have surae (odg f they do ad 0 f they do ot) ad aordg to ther odto, whh s rated as good (g), far (f), or serous (s). Cosder a expermet that ossts of the odg of suh a patet. (a) Gve the sample spae of ths expermet. (b) Let A be the evet that the patet s serous odto. Spefy the outomes A. () Let B be the evet that the patet s usured. Spefy the outomes B. (d) Gve all the outomes the evet B»A T3. F=FE»FE, ad E»F=E»E F.

2 T5. For ay sequee of evets E, E 2,, defe a ew sequee F, F 2, of dsot evets (that s, evets suh that F F =«wheever ) suh that for all, U U F = E (Note: we a prove the very useful Boole s equalty: P U A A ) by applyg = = the result of ths problem.) T6. Let E, F, ad G be three evets. Fd expressos for the evets so that of E, F, ad G: (a) oly E ours; (b) both E ad G but ot F our; () at least oe of the evets ours; (d) at least two of the evets our; (e) all three our; T7. Fd the smplest expresso for the followg evets: (a) (E» F)(E» F ) (b) (E» F)(E» F)(E» F ); () (E» F)(F» G). Set 2: P5. For years, telephoe area odes the Uted States ad Caada ossted of a sequee of three dgts. The frst dgt was a teger betwee 2 ad 9; the seod dgt was ether 0 or ; the thrd dgt was ay teger betwee ad 9. How may area odes were possble? How may area odes startg wth a 4 were possble? P7. (a) I how may ways a 3 boys ad 3 grls st a row? (b) I how may ways a 3 boys ad 3 grls st a row f the boys ad the grls are eah to st together? () I how may ways f oly the boys must st together? (d) I how may ways f o two people of the same sex are allowed to st together? P3. Cosder a group of 20 people. If everyoe shakes hads wth everyoe else, how may hadshakes take plae? P5. A dae lass ossts of 22 studets, 0 wome ad 2 me. If 5 me ad 5 wome are to be hose ad the pared off, how may results are possble? P2. Cosder the grd of pots show below. Suppose that startg at the pot labeled A

3 you a go oe step up or oe step to the rght at eah move. Ths s otued utl the pot labeled B s reahed. How may dfferet paths from A to B are possble? Ht: Note that to reah B from A you must take 4 steps to the rght ad 3 steps upward. B A P22. I problem 2, how may dfferet paths are there from A to B that go through the pot rled below? + m m m m T8. Prove that = + + L + r 0 r r r 0 Ht: Cosder a group of me ad m wome. How may groups of sze r are possble? 2 T9. Use Theoretal Exerse 8 to prove that = k = 0 k 2 T2. Cosder the followg ombatoral detty: k = 2 k = k (a) Preset a ombatoral argumet for the above by osderg a set of people ad determg, two ways, the umber of possble seletos of a ommttee of ay sze ad a harperso for the ommttee. Ht: () How may possble seletos are there of a ommttee of sze k ad ts har perso? () How may possble seletos are there of a harperso ad the other ommttee members? 2. Aswerg questos of Ch ad Ch2: 3. Codtoal Probablty: Defto: For evets E ad F, f F)>0, the the odtoal probablty of E gve that F EF) has ourred s deoted by ad s defed by E F) = F) The multplato rule: E E 2 E 3 E ) = E ) E 2 E ) E E E 2 E- )

4 4. Bayes Formula: A useful detty s that E) = E F) F) + E F ) F ). It a be used to ompute E) by odtog o whether F ours. H) / H ) s alled the odds of the evet H. The odd of a evet H tells how muh more lkely t s that the evet H ours tha t s that t does ot our. For stae, f H) = 2/3, the H) = 2, so the odds s 2. If the odds s equal to t, the t s ommo to say that the H E) H ) E H ) odds are t to favor of the hypothess. The detty = H E) H ) E H ) shows that whe ew evdee E s obtaed, the value of the odds rato of H beomes ts old value multpled by the rato of the odtoal probablty of the ew evdee whe H s true to ts odtoal probablty whe H s ot true. The detty E F ) F ) F ) = s kow as Bayes formula. If the evets F, = E F ) F ) =,,, are ompetg hypotheses, the Bayes formula shows how to ompute the odtoal probabltes of these hypotheses whe addtoal evdee E beomes avalable. 5. Idepedee: Defto: Two evets E ad F are sad to be depedet f EF) = E)F). If E)>0 or F)>0, the odto s equvalet to F E) = F) (for E)>0) ad to E F) = E) (for F)>0). Thus E ad F are depedet f kowledge of the ourree of oe of them does ot affet the probablty of the other. Two evets E ad F that are ot depedet are sad to be depedet. If E ad F are depedet, the so are E ad F. The evets E,, E are sad to be depedet f for ay subset E,, E r of them, E E r ) = E ) E r ). For a fxed evet F, E F) a be osdered to be a probablty futo o the evets E of the sample spae. 6. Problems to be dsussed: Problems: P2. If two far de are rolled, what s the odtoal probablty that the frst oe lads o 6 gve that the sum of the de s? Compute for all values of betwee 2 ad 2. P8. A ouple has 2 hldre. What s the probablty that both are grls f the eldest s a grl?

5 P8. A total of 46 peret of the voters a erta ty lassfy themselves as Idepedets, whereas 30 peret lassfy themselves as Lberals ad 24 peret as Coservatves. I a reet loal eleto, 35 peret of the Idepedets, 62 peret of the Lberals, ad 58 peret of the Coservatves voted. A voter s hose at radom. Gve that ths perso voted the loal eleto, what s the probablty that he or she s (a) a Idepedet; (b) a Lberal; () a Coservatve? (d) What frato of voters partpated the loal eleto? P5. A worker has asked her supervsor for a letter of reommedato for a ew ob. She estmates that there s a 80 peret hae that she wll get the ob f she reeves a strog reommedato, a 40 peret hae f she reeves a moderately good reommedato, ad a 0 peret hae f she reeves a weak reommedato. She further estmates that the probabltes that the reommedato wll be strog, moderate, or weak are.7,.2, ad., respetvely. (a) How erta s she that she wll reeve the ew ob offer? (b) Gve that she does reeve the offer, how lkely should she feel that she reeved a strog reommedato; a moderate reommedato; a weak reommedato? () Gve that she does ot reeve the ob offer, how lkely should she feel that she reeved a strog reommedato; a moderate reommedato; a weak reommedato?