STAT 400 Discussio 0 Solutios Spig 08. ~.5 ~.6 At the begiig of a cetai study of a goup of pesos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the fiveyea study, it was detemied that the death ates of the heavy smoes ad light smoes wee seve ad thee times that of the osmoes, espectively. A adomly selected paticipat died ove the fiveyea peiod. What is the pobability that the paticipat was a osmoe? A light smoe? A heavy smoe? P ( H ) 0.5, P ( H ) 0.30, P ( N ) 0.55, P ( D H ) 7 p, P ( D L ) 3 p, P ( D N ) p. P ( D ) P ( H ) P ( D H ) + P ( L ) P ( D L ) + P ( N ) P ( D N ) 0.5 7 p + 0.30 3 p + 0.55 p.05 p + 0.90 p + 0.55 p.50 p..05 p P ( H D ).50 p 0.4. 0.90 p P ( L D ).50 p 0.36. 0.55 p P ( N D ).50 p 0..
. At a hospital s emegecy oom, patiets ae classified ad 0% of them ae citical, 30% ae seious, ad 50% ae stable. Of the citical oes, 30% die; of the seious, 0% die; ad of the stable, % die. a) ~.55 ~.65 ~ Give that a patiet dies, what is the coditioal pobability that the patiet was classified as citical? As seious? As stable? P ( patiet dies ) 0.0 0.30 + 0.30 0.0 + 0.50 0.0 0.06 + 0.03 + 0.0 0.0. 0.06 P ( citical patiet dies ) 0.0 0.60. 0.03 P ( seious patiet dies ) 0.0 0.30. 0.0 P ( stable patiet dies ) 0.0 0.0. b) Ae evets {a patiet dies} ad {a patiet was classified as citical} idepedet? Justify you aswe. P ( patiet dies Ç citical ) 0.06 ¹ 0.0 0.0 P ( patiet dies ) P ( citical ). P ( patiet dies citical ) 0.30 ¹ 0.0 P ( patiet dies ). P ( citical patiet dies ) 0.60 ¹ 0.0 P ( citical ). {a patiet dies} ad {a patiet was classified as citical} ae NOT idepedet. c) Ae evets {a patiet dies} ad {a patiet was classified as seious} idepedet? Justify you aswe. P ( patiet dies Ç seious ) 0.03 0.0 0.30 P ( patiet dies ) P ( seious ). P ( patiet dies seious ) 0.0 0.0 P ( patiet dies ). P ( seious patiet dies ) 0.30 0.30 P ( seious ). {a patiet dies} ad {a patiet was classified as citical} ae idepedet.
3. Alex leas that his favoite socce team, UbaaChampaig Uited ( ÈÌÈ ), has a 70% chace of sigig oe of the best playes i the wold, Ro Aldo. He immediately us some compute simulatios ad discoves that if ÈÌÈ sigs Ro Aldo, it would have a 0.90 pobability of wiig the Ameica Cetal Illiois Divisio champioship. Ufotuately, if ÈÌÈ does ot sig Ro Aldo, the the pobability of wiig the champioship is oly 0.40. Alex becomes too excited ad slips ito a coma. He comes out of the coma a yea late ad fids out that ÈÌÈ has wo the champioship. What is the pobability that ÈÌÈ was able to sig Ro Aldo? P ( RA ) 0.70, P ( W RA ) 0.90, P ( W RA' ) 0.40. Bayes Theoem: P ( RA W ) P( RA ) P( P( RA ) W RA ) P( W RA ) + P( RA' ) P( W RA 0.70 0.90 0.63 0.63 0.84. 0.70 0.90 + 0.30 0.40 0.63 + 0. 0. 75 ' ) OR RA RA' W 0.90 0.70 0.63 0.40 0.30 0. W' 0.07 0.70 0.8 0.30 0.75 0.5.00 0.63 P ( RA W ) 0.84. 0.75
4. Jac, Mie ad Tom ae oommates, ad evey Suday ight they split a lage pizza fo die. Whe thee is oly oe slice left, the pobability that Jac wats it is 0.40, the pobability that Mie wats it is 0.35, ad the pobability that Tom wats it is 0.5. Suppose that whethe o ot each oe of them will wat the last slice is idepedet of the othe two. ( Jac ) 0.40, P( Jac' ) 0.60, ( Mie ) 0.35, P( Mie' ) 0.65, ( Tom ) 0.5, P( Tom' ) 0.75. a) What is the pobability that oly oe of the oommates will wat the last slice? oly Jac Jac Mie' Tom' 0.40 0.65 0.75 0.950, oly Mie Jac' Mie Tom' 0.60 0.35 0.75 0.575, oly Tom Jac' Mie' Tom 0.60 0.65 0.5 0.0975. P( oly oe wats the last slice ) P( oly Jac o oly Mie o oly Tom ) P( oly Jac ) + P( oly Mie ) + P( oly Tom ) 0.950 + 0.575 + 0.0975 0.45. b) What is the pobability that at least oe of the oommates will wat the last slice? P( at least oe wats the last slice ) P( o oe wats the last slice ) P( Jac' Ç Mie' Ç Tom' ) 0.60 0.65 0.75 0.95 0.7075. c) What is the pobability that at most oe of the oommates will wat the last slice? P( at most oe wats the last slice ) P( oly oe wats the last slice ) + P( o oe wats the last slice ) 0.45 + 0.95 0.745.
5. Dug A is effective with pobability 0.80. Dug B is effective with pobability 0.70. Thee is a 40% chace of a egative dug iteactio betwee dugs A ad B. Suppose the effectiveess of the two dugs ad the possibility of a egative dug iteactio ae all idepedet. Fid the pobability that a) both dugs ae effective ad thee is o egative dug iteactio. ( Dug A is effective ) AND ( Dug B is effective ) AND ( o egative dug iteactio ) 0.80 0.70 0.60 0.336. b) at least oe of the two dugs is effective ad thee is o egative dug iteactio. [ 0.80 + 0.70 0.80 0.70 ] 0.60 0.564. OR [ ( 0.80 ) ( 0.70 ) ] 0.60 0.564. OR 0.80 0.70 0.60 + 0.80 0.30 0.60 + 0.0 0.70 0.60 0.336 + 0.44 + 0.084 0.564. c) at most oe of the two dugs is effective ad thee is egative dug iteactio. [ 0.80 0.70 ] 0.40 0.76. OR 0.0 0.30 0.40 + 0.80 0.30 0.40 + 0.0 0.70 0.40 0.04 + 0.096 + 0.056 0.76.
6. A ba has two emegecy souces of powe fo its computes. Thee is a 95% chace that souce will opeate duig a total powe failue, ad a 80% chace that souce will opeate. Assume the powe souces ae idepedet. What is the pobability that at least oe of them will opeate duig a total powe failue? 0.95 + 0.80 0.95 0.80 0.95 + 0.80 0.76 0.99. OR 0.95 + 0.05 0.80 0.99. OR 0.05 0.0 0.99. 7..4.5 If P ( A ) 0.3, P ( B ) 0.6. a) Fid P ( A È B ) whe A ad B ae idepedet. P ( A Ç B ) P ( A ) P ( B ) 0.3 0.6 0.8; P ( A È B ) P ( A ) + P ( B ) P ( A Ç B ) 0.3 + 0.6 0.8 0.7. b) Fid P ( A B ) whe A ad B ae mutually exclusive. P ( A Ç B ) 0 P ( A Ç B ) 0; P ( A B ) 0. P( B ) 0. 6
8..45.55 If P ( A ) 0.8, P ( B ) 0.5, ad P ( A È B ) 0.9, ae A ad B idepedet evets? Why o why ot? P ( A È B ) P ( A ) + P ( B ) P ( A Ç B ). 0.90 0.80 + 0.50 P ( A Ç B ). P ( A Ç B ) 0.40 0.80 0.50 P ( A ) P ( B ). Þ P ( A Ç B ) 0.40. Þ A ad B ae idepedet. 9..48.58 Die A has oage o oe face ad blue o five faces, Die B has oage o two faces ad blue o fou faces, Die C has oage o thee faces ad blue o thee faces. All ae fai dice. If the thee dice ae olled, fid the pobability that exactly two of the thee dice come up oage? 3 4 3 5 3 P ( O O B ) + P ( O B O ) + P ( B O O ) + +. 6 6 6 6 6 6 6 6 6 9
0. A electoic device has fou idepedet compoets. Two of those fou ae ew, ad have a eliability of 0.80 each, oe is old, with 0.75 eliability, ad oe is vey old, ad its eliability is 0.60. Let Ai { i th compoet is fuctioal }. The P( A ) P( A ) 0.80, P( A3 ) 0.75, P( A4 ) 0.60. a) Suppose that the device wos if all fou compoets ae fuctioal. What is the pobability that the device will wo whe eeded? all fou st ad d ad 3d ad 4th itesectio. P( A Ç A Ç A3 Ç A4 ) sice the compoets ae idepedet P( A ) P( A ) P( A3 ) P( A4 ) (0.80) (0.80) (0.75) (0.60) 0.88. b) Suppose that the device wos if at least oe of the fou compoets is fuctioal. What is the pobability that the device will wo whe eeded? at least oe eithe st o d o 3d o 4th o 5th uio. P(at least oe) P(oe). oe ot st ad ot d ad ot 3d ad ot 4th ad ot 5th. P( A È A È A3 È A4 ) P( A' Ç A' Ç A3' Ç A4' ) sice the compaies ae idepedet P( A' ) P( A' ) P( A3' ) P( A4' ) (0.0) (0.0) (0.5) (0.40) 0.004 0.996.
c) Suppose that the fou compoets ae coected as show o the diagam below. Fid the eliability of the system. 0.60 0.80 0.80 0.75 0.60 0.80 0.75 0.80 0.80 0.75 0.60. 0.60 0.60 0.80 0.40 0.40 0.84. OR 0.60 + 0.60 0.60 0.60 0.84. 0.84 0.80 0.84 0.80 0.67.
. A oal fial exam cotiues util a studet eithe aswes two questios i a ow coectly (ad passes) o aswes two questios i a ow icoectly (ad fails). Suppose Alex has pobability p to aswe ay questio coectly, idepedetly of ay othe questios. What is the pobability that Alex would pass the exam? C C C W C C C W C W C C S ( C W ) C C p p ( p ) W C C W C W C C W C W C W C C S ( W C ) W C C ( p ) p ( p ) p p p ( p ) ( p ) p p ( p ) p p p p +.
. Suppose Jae has a fai 4sided die, ad Dic has a fai 6sided die. Each day, they oll thei dice at the same time (idepedetly) util someoe olls a (as may times as ecessay). (The the peso who did ot oll a does the dishes.) Fid the pobability that Jae olls the fist befoe Dic does. ( J D' ) o ( J' D' ) ( J D' ) o ( J' D' ) ( J' D' ) ( J D' ) o 5 4 6 + 3 5 5 4 6 4 6 + 3 5 3 5 5 4 6 4 6 4 6 +... å 5 3 5 5 4 6 5. 0 4 6 4 6 3 5 9 4 6 OR tu D D' 5 3 + + 4 4 4 5 5 game eds J 4 6 4 4 6 4 o oe does dishes Dic does dishes 3 3 3 5 5 5 9 J' 4 6 4 4 6 4 4 4 Jae does dishes game cotiues Jae wis 5 9 9 4
3. Pove (show) that a).0.34 ( Pascal s equatio ).. b).. + + + ú û ù ê ë é + ú û ù ê ë é
4. We aleady ow that. Pove (show) that.. å 0 å 0 å 0 å å å å å 0 m m m å 0 m m
5..4.36 The eatig club is hostig a maeyouow sudae paty at which the followig ae povided: Ice Ceam Flavos Chocolate Cooies ceam Stawbey Vailla Toppigs Caamel Hot fudge Mashmallow M&M s Nuts Stawbeies a) How may sudaes ae possible usig oe flavo of ice ceam ad thee diffeet toppigs? 6 4 80. 3 b) How may sudaes ae possible usig oe flavo of ice ceam ad fom zeo to six toppigs? 4 6 56. c) How may diffeet combiatios of flavos of thee scoops of ice ceam ae possible if it is pemissible to mae all thee scoops the same flavo? The umbe of uodeed selectios of 3 objects that ca be made out of 4 objects 4 + 3 6 (epetitios ae allowed) is 0. 3 3 3 4 4 4 3 3 3 3 3 3 3 4 3 3 4 4 3 4 3 3 3 4 4 4 4 3 4 4 4 4