STAT 400 Wee 03 Discussio Fall 07. ~.5- ~.6- At the begiig of a cetai study of a gou of esos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the five-yea study, it was detemied that the death ates of the heavy smoes ad light smoes wee seve ad thee times that of the osmoes, esectively. A adomly selected aticiat died ove the five-yea eiod. What is the obability that the aticiat was a osmoe? A light smoe? A heavy smoe?. At a hosital s emegecy oom, atiets 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) ~.5-5 ~.6-5 ~ Give that a atiet dies, what is the coditioal obability that the atiet was classified as citical? As seious? As stable? b) Ae evets {a atiet dies} ad {a atiet was classified as citical} ideedet? Justify you aswe. c) Ae evets {a atiet dies} ad {a atiet was classified as seious} ideedet? Justify you aswe. 3. Alex leas that his favoite socce team, Ubaa-Chamaig Uited ( ), has a 70% chace of sigig oe of the best layes i the wold, Ro Aldo. He immediately us some comute simulatios ad discoves that if sigs Ro Aldo, it would have a 0.90 obability of wiig the Ameica Cetal Illiois Divisio chamioshi. Ufotuately, if does ot sig Ro Aldo, the the obability of wiig the chamioshi is oly 0.40. Alex becomes too excited ad slis ito a coma. He comes out of the coma a yea late ad fids out that has wo the chamioshi. What is the obability that was able to sig Ro Aldo?
4. Jac, Mie ad Tom ae oommates, ad evey Suday ight they slit a lage izza fo die. Whe thee is oly oe slice left, the obability that Jac wats it is 0.40, the obability that Mie wats it is 0.35, ad the obability that Tom wats it is 0.5. Suose that whethe o ot each oe of them will wat the last slice is ideedet of the othe two. a) What is the obability that oly oe of the oommates will wat the last slice? b) What is the obability that at least oe of the oommates will wat the last slice? c) What is the obability that at most oe of the oommates will wat the last slice? 5. Dug A is effective with obability 0.80. Dug B is effective with obability 0.70. Thee is a 40% chace of a egative dug iteactio betwee dugs A ad B. Suose the effectiveess of the two dugs ad the ossibility of a egative dug iteactio ae all ideedet. Fid the obability that a) both dugs ae effective ad thee is o egative dug iteactio. b) at least oe of the two dugs is effective ad thee is o egative dug iteactio. c) at most oe of the two dugs is effective ad thee is egative dug iteactio. 6. A ba has two emegecy souces of owe fo its comutes. Thee is a 95% chace that souce will oeate duig a total owe failue, ad a 80% chace that souce will oeate. Assume the owe souces ae ideedet. What is the obability that at least oe of them will oeate duig a total owe failue? 7..4-.5- If P ( A ) 0.3, P ( B ) 0.6. a) Fid P ( A B ) whe A ad B ae ideedet. b) Fid P ( A B ) whe A ad B ae mutually exclusive.
8..4-5.5-5 If P ( A ) 0.8, P ( B ) 0.5, ad P ( A B ) 0.9, ae A ad B ideedet evets? Why o why ot? 9..4-8.5-8 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 obability that exactly two of the thee dice come u oage? 0. A electoic device has fou ideedet comoets. 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. a) Suose that the device wos if all fou comoets ae fuctioal. What is the obability that the device will wo whe eeded? b) Suose that the device wos if at least oe of the fou comoets is fuctioal. What is the obability that the device will wo whe eeded? c) Suose that the fou comoets ae coected as show o the diagam below. Fid the eliability of the system. 0.60 0.80 0.75 0.80. A oal fial exam cotiues util a studet eithe aswes two questios i a ow coectly (ad asses) o aswes two questios i a ow icoectly (ad fails). Suose Alex has obability to aswe ay questio coectly, ideedetly of ay othe questios. What is the obability that Alex would ass the exam?
. Suose Jae has a fai 4-sided die, ad Dic has a fai 6-sided die. Each day, they oll thei dice at the same time (ideedetly) util someoe olls a (as may times as ecessay). (The the eso who did ot oll a does the dishes.) Fid the obability that Jae olls the fist befoe Dic does. 3. Pove (show) that a).-0.3-4 b) ( Pascal s equatio ).. 4. We aleady ow that 0. Pove (show) that 0. 5..-4.3-6 The eatig club is hostig a mae-you-ow sudae aty at which the followig ae ovided: Ice Ceam Flavos Chocolate Cooies ceam Stawbey Vailla Toigs Caamel Hot fudge Mashmallow M&M s Nuts Stawbeies a) How may sudaes ae ossible usig oe flavo of ice ceam ad thee diffeet toigs? b) How may sudaes ae ossible usig oe flavo of ice ceam ad fom zeo to six toigs? c) How may diffeet combiatios of flavos of thee scoos of ice ceam ae ossible if it is emissible to mae all thee scoos the same flavo?
Aswes:. ~.5- ~.6- At the begiig of a cetai study of a gou of esos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the five-yea study, it was detemied that the death ates of the heavy smoes ad light smoes wee seve ad thee times that of the osmoes, esectively. A adomly selected aticiat died ove the five-yea eiod. What is the obability that the aticiat 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 ( D L ) 3, P ( D N ). P ( D ) P ( H ) P ( D H ) + P ( L ) P ( D L ) + P ( N ) P ( D N ) 0.5 7 + 0.30 3 + 0.55.05 + 0.90 + 0.55.50. P ( H D ) P ( L D ) P ( N D ).05.50 0.90.50 0.55.50 0.4. 0.36. 0..
. At a hosital s emegecy oom, atiets 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) ~.5-5 ~.6-5 ~ Give that a atiet dies, what is the coditioal obability that the atiet was classified as citical? As seious? As stable? P ( atiet dies ) 0.0 0.30 + 0.30 0.0 + 0.50 0.0 0.06 + 0.03 + 0.0 0.0. P ( citical atiet dies ) P ( seious atiet dies ) P ( stable atiet dies ) 0.06 0.0 0.03 0.0 0.0 0.0 0.60. 0.30. 0.0. b) Ae evets {a atiet dies} ad {a atiet was classified as citical} ideedet? Justify you aswe. P ( atiet dies citical ) 0.06 0.0 0.0 P ( atiet dies ) P ( citical ). P ( atiet dies citical ) 0.30 0.0 P ( atiet dies ). P ( citical atiet dies ) 0.60 0.0 P ( citical ). {a atiet dies} ad {a atiet was classified as citical} ae NOT ideedet. c) Ae evets {a atiet dies} ad {a atiet was classified as seious} ideedet? Justify you aswe. P ( atiet dies seious ) 0.03 0.0 0.30 P ( atiet dies ) P ( seious ). P ( atiet dies seious ) 0.0 0.0 P ( atiet dies ). P ( seious atiet dies ) 0.30 0.30 P ( seious ). {a atiet dies} ad {a atiet was classified as citical} ae ideedet.
3. Alex leas that his favoite socce team, Ubaa-Chamaig Uited ( ), has a 70% chace of sigig oe of the best layes i the wold, Ro Aldo. He immediately us some comute simulatios ad discoves that if sigs Ro Aldo, it would have a 0.90 obability of wiig the Ameica Cetal Illiois Divisio chamioshi. Ufotuately, if does ot sig Ro Aldo, the the obability of wiig the chamioshi is oly 0.40. Alex becomes too excited ad slis ito a coma. He comes out of the coma a yea late ad fids out that has wo the chamioshi. What is the obability that 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( W RA ) P( RA ) P( W RA ) P( RA' ) P( W RA' ) 0.70 0.90 0.700.90 0.300.40 0.63 0.63 0. 0.63 0.75 0.84. OR W W' RA RA' 0.90 0.70 0.63 0.40 0.30 0. 0.07 0.70 0.8 0.30 0.75 0.5.00 P ( RA W ) 0.63 0.75 0.84.
4. Jac, Mie ad Tom ae oommates, ad evey Suday ight they slit a lage izza fo die. Whe thee is oly oe slice left, the obability that Jac wats it is 0.40, the obability that Mie wats it is 0.35, ad the obability that Tom wats it is 0.5. Suose that whethe o ot each oe of them will wat the last slice is ideedet 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 obability 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 obability 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' Mie' Tom' ) 0.60 0.65 0.75 0.95 0.7075. c) What is the obability 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 obability 0.80. Dug B is effective with obability 0.70. Thee is a 40% chace of a egative dug iteactio betwee dugs A ad B. Suose the effectiveess of the two dugs ad the ossibility of a egative dug iteactio ae all ideedet. Fid the obability 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 owe fo its comutes. Thee is a 95% chace that souce will oeate duig a total owe failue, ad a 80% chace that souce will oeate. Assume the owe souces ae ideedet. What is the obability that at least oe of them will oeate duig a total owe 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 B ) whe A ad B ae ideedet. P ( A B ) P ( A ) P ( B ) 0.3 0.6 0.8; P ( A B ) P ( A ) + P ( B ) P ( A B ) 0.3 + 0.6 0.8 0.7. b) Fid P ( A B ) whe A ad B ae mutually exclusive. P ( A B ) 0; P ( A B ) P A P B 0 B 0. 6 0.
8..4-5.5-5 If P ( A ) 0.8, P ( B ) 0.5, ad P ( A B ) 0.9, ae A ad B ideedet evets? Why o why ot? P ( A B ) P ( A ) + P ( B ) P ( A B ). 0.90 0.80 + 0.50 P ( A B ). P ( A B ) 0.40 0.80 0.50 P ( A ) P ( B ). P ( A B ) 0.40. A ad B ae ideedet. 9..4-8.5-8 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 obability that exactly two of the thee dice come u oage? P ( O O B ) + P ( O B O ) + P ( B O O ) 6 6 3 6 6 4 6 3 6 5 6 6 3. 6 9
0. A electoic device has fou ideedet comoets. 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 comoet is fuctioal }. The P( A ) P( A ) 0.80, P( A3 ) 0.75, P( A4 ) 0.60. a) Suose that the device wos if all fou comoets ae fuctioal. What is the obability that the device will wo whe eeded? all fou st ad d ad 3d ad 4th itesectio. P( A A A3 A4 ) sice the comoets ae ideedet P( A ) P( A ) P( A3 ) P( A4 ) (0.80) (0.80) (0.75) (0.60) 0.88. b) Suose that the device wos if at least oe of the fou comoets is fuctioal. What is the obability 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 A A3 A4 ) P( A' A' A3' A4' ) sice the comaies ae ideedet P( A' ) P( A' ) P( A3' ) P( A4' ) (0.0) (0.0) (0.5) (0.40) 0.004 0.996.
c) Suose that the fou comoets 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 asses) o aswes two questios i a ow icoectly (ad fails). Suose Alex has obability to aswe ay questio coectly, ideedetly of ay othe questios. What is the obability that Alex would ass the exam? C C C W C C C W C W C C ( C W ) C C W C C W C W C C W C W C W C C ( W C ) W C C +.
. Suose Jae has a fai 4-sided die, ad Dic has a fai 6-sided die. Each day, they oll thei dice at the same time (ideedetly) util someoe olls a (as may times as ecessay). (The the eso who did ot oll a does the dishes.) Fid the obability 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... 3 5 5 0 4 6 4 6 5 4 6 3 5 4 6 9 5. OR tu D D' 5 5 J 4 6 4 4 6 4 o oe does dishes Dic does dishes 3 3 3 5 5 J' 4 6 4 4 6 4 Jae does dishes game cotiues 4 5 3 9 4 4 4 game eds 5 9 4 4 Jae wis 9 5
3. Pove (show) that a).-0.3-4 ( Pascal s equatio ).. b)..
4. We aleady ow that 0. Pove (show) that 0. 0 0 m m m 0 m m.
5..-4.3-6 The eatig club is hostig a mae-you-ow sudae aty at which the followig ae ovided: Ice Ceam Flavos Chocolate Cooies ceam Stawbey Vailla Toigs Caamel Hot fudge Mashmallow M&M s Nuts Stawbeies a) How may sudaes ae ossible usig oe flavo of ice ceam ad thee diffeet toigs? 6 4 3 80. b) How may sudaes ae ossible usig oe flavo of ice ceam ad fom zeo to six toigs? 4 6 56. c) How may diffeet combiatios of flavos of thee scoos of ice ceam ae ossible if it is emissible to mae all thee scoos the same flavo? The umbe of uodeed selectios of 3 objects that ca be made out of 4 objects 4 3 6 (eetitios ae allowed) is 0. 3 3 3 44 4 3 333 3 33 4 334 4 34 33 344 44 34 444