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M@0humoC//Z;v+/:h^;)JS~K;-uK2Miߩ~mKz+y3|]_v)$~~0旵>B9+!/M#$6莥z?+ƖebTl_W _yS ãkҳ(?~?l(dswNxX\a3&;9< 6(h|) :RcExk9DqV|"~3|s4~a*u[oGM)J+I-L.M0K1K4J5K7J8K9J: K< J=K3 J? J@LAIKIBKCDE$$$"$BCVo{)ü$"$=)~ |?)$$"$'"M0c!c= 0AA@ʚ;n.ʚ;g44d4d 0ppp@ <4!d!d8A10l<4dddd8A10l <4BdBd8c20l80___PPT10 ?  %"Deal or No Deal? Fair or Not Fair?=Dr. Jason Gershman Rice University School Mathematics ProjectDeal or No Deal?Game show Choose 1 of 26 cases containing $0.01 - $1 million $>ZZ@The Game Board "The Game Board Again  ;Mean: $131477 (x with a bar on top) Median: $875 (Med) JZ;#The Game Begins8First, the contestant chooses his case and this is put aside, sealed. He then opens cases that he didn t choose. The first case he opens contains $10000. $Game Board at 1st Deal$ ^.Game Board at 1st Deal$  Mean: $95405 Median: $875 *Game Board at 2nd Deal$ _/Game Board at 2nd Deal$  Mean: $112097 Median: $1000 -Game Board at 3rd Deal$ b0Game Board at 3rd Deal$ d1Game Board at 3rd Deal$  Mean: $139195 Median: $1000  Fairness Offer : Mean Ratio(First Deal : 11000/95405 = 12% of Mean Second Deal: 24000/112097= 21% of Mean Third Deal: 50000/139195 = 36% of Mean Why are the deals so  unfair Why do you think NBC is willing to pay a greater percentage of the mean now than before?vwf2Game Board at 4th Deal  What is the average of the 3 amounts that he removed ($300000, $5000 and $25?) Is this above or below the previous mean value of $139195? g3Game Board at 4th Deal  Mean: $153266 Median: $537.50Z|<Game Board at 4th Deal  UTwo more cases must be eliminated. How many ways can the contestant select two cases?VVj4Game Board at 5th Deal 9 Observationsl5Game Board at 5th Deal  Mean: $66855 Median: $62.50Zp6Game Board at 6th Deal q7Game Board at 6th Deal  Mean: $80226 Median: $75Zr8Game Board at 7th Deal s9Game Board at 7th Deal  Mean: $100033 Median: $62.50Z@  Some Thoughtsx:Game Board at 8th Deal y;Game Board at 8th Deal  Mean: $133360 Median: $75ZCFairnessC" Conclusions4What do you think of the contestant s decisions? Does it really matter which case the contestant chooses at the start of the game? Comments / Questions ~=New Game Board  The top prize changes from $1 million to $2 million while the other 25 values remain the same. How does this change the mean? The median? Does changing this one value affect the expected outcome in the game?Z>New Game Board  Original Mean = $131477 New Mean = $169937 New Mean is a 29% increase over the Old Mean Old Median = $875 New Median = $875 Changing 4% of the values changes the median by 0%.|.Z-ZZ&Z4Z.-&4?Premier Game Board  yNBC doubled all 7 top values during the premier episode. How did this change the mean? How did this change the median? d:Z>ZZZ:>  APremier Game Board  Original Mean = $131477 New Mean = $256477 New Mean is a 95% increase over the Old Mean Old Median = $875 New Median = $875 Changing 27% of the values changes the median by 0%.|.Z-ZZ&Z5Z.-&5BAlternative Game Board  Original Mean = $131477 New Mean = $131483 New Mean is a $6 increase over the Old Mean Old Median = $875 New Median = $875d.Z,ZZ&Z.,&  /% &(,.0=GJ!`,a-c.e/h0i1k2m3t4u5v6w7z8{9}:;<=@AB 0` ` ̙33` 333MMM` ff3333f` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> $(    6h P  T Click to edit Master title style! !  0T   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0  ``  X*  0X `   Z*  08 `   Z*H  0޽h ? ̙33 Default Design0 zrPD (  D D 0TC *   P*   D 0LH  *  R*  d D c $ ?v   D 0`K    RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S D 60 ?   P*   D 64  ?  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P 6`B $ 01 ? `B % 01 ?L L `B & 01 ?2 2`B ' 01 ? `B ( 01 ?f f `B ) 01 ? `B * 01 ? `B + 01 ? `B , 01 ? `B - 01 ? `B . 01 ? fB / 6o ?,P PfB 0 6o ?6fB 1 6o ? 6 fB 2 6o ?fB 3 6o ?  fB 4 6o ?fB 5 6o ?  fB 6 6o ?fB 7 6o ?  fB 8 6o ? fB 9 6o ?  fB : 6o ? fB ; 6o ? fB < 6o ? fB = 6o ? fB > 6o ? f fB ? 6o ? f fB @ 6o ?f L fB A 6o ? f L fB B 6o ?L 2fB C 6o ? L 2fB D 6o ?2fB E 6o ? 2 fB F 6o ?fB G 6o ?   I 0N  `,$0 n&The 1st offer from the bank is $11000.,' 2   J <Q   <$@0    L 0T 0  ,$0 8What do you think the 1st offer from the bank should be?,9   H  0޽h ? ̙33$  ___PPT10 .g4aw+A9D` ' ` = @B D ' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*J %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*J %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I%(+p+0+I0 ++0+L0 +* 0  **HN)(  r  S p P   ' P  # #"F: P@   <  ?,  S100 000 @`   <T  ? ,  S 75n @`   <D  ?,  R75 000 @`   <x  ?,  S 50n @`   <  ?,  R50 000 @`   <P  ?, S 25n @`  <  ?,  V25 000 @`  <  ?, c 10(n @`  <$Ԥ  ?,  V10 000 @`  <֤  ?, R 5n @`  <ߤ  ?,6  Q5 000 @`  <`  ?6, d 1(nn @`  <|  ?, f  S300 000 @`  <  ? ,f  T 200n @`  <  ?,  W200 000 @`  <  ? ,  T 100n @`  <l  ?,L 2 W500 000 @`  <("  ?L ,2 d 400(n @`  <3  ?,f L  S400 000 @`  <h5  ?f ,L  d 300(n @`  <HF  ?,2  S750 000 @`  <H  ?2, d 500(n @`  <Y  ?,  Y 1 000 000   @`  <X\  ?, X 750n @`   <l  ?,P 6 S1 000 @` ! <u  ?P,6 S.01  hh@`fB " 6o ?P,P`B # 01 ?6 6`B $ 01 ? fB % 6o ?P6`B & 01 ?,P,fB ' 6o ? P 6`B ( 01 ? `B ) 01 ?L L `B * 01 ?2 2`B + 01 ? `B , 01 ?f f `B - 01 ? `B . 01 ? `B / 01 ? `B 0 01 ? `B 1 01 ? `B 2 01 ? fB 3 6o ?,P PfB 4 6o ?6fB 5 6o ? 6 fB 6 6o ?fB 7 6o ?  fB 8 6o ?fB 9 6o ?  fB : 6o ?fB ; 6o ?  fB < 6o ? fB = 6o ?  fB > 6o ? fB ? 6o ? fB @ 6o ? fB A 6o ? fB B 6o ? f fB C 6o ? f fB D 6o ?f L fB E 6o ? f L fB F 6o ?L 2fB G 6o ? L 2fB H 6o ?2fB I 6o ? 2 fB J 6o ?fB K 6o ?   N 6    \Data from 1st Deal: Mean= $95405 Median = $875 The first offer from the bank was $11000.*]  P%H  0޽h ? ̙33___PPT10i.i4I +D=' ` = @B +2 -0 **II *(  x  c $xc P   ' P  # #"F: P@   <  ?,  S100 000 @`  <P  ? ,  S 75n @`  <  ?,  R75 000 @`  <  ?,  S 50n @`  <ҥ  ?,  R50 000 @`   <\ե  ?, S 25n @`   <  ?,  V25 000 @`   <ߥ  ?, c 10(n @`   <  ?,  V10 000 @`   <  ?, R 5n @`  <  ?,6  Q5 000 @`  <l  ?6, d 1(nn @`  <  ?, f  S300 000 @`  <|!  ? ,f  T 200n @`  <2  ?,  W200 000 @`  <-  ? ,  T 100n @`  <@D  ?,L 2 W500 000 @`  <F  ?L ,2 X 400n @`  < W  ?,f L  S400 000 @`  <P  ?f ,L  d 300(n @`  <i  ?,2  S750 000 @`  <pl  ?2, X 500n @`  <|  ?,  Y 1 000 000   @`  <v  ?, X 750n @`  <  ?,P 6 S1 000 @`  <l  ?P,6 S.01  hh@`fB  6o ?P,P`B  01 ?6 6`B   01 ? fB ! 6o ?P6`B " 01 ?,P,fB # 6o ? P 6`B $ 01 ? `B % 01 ?L L `B & 01 ?2 2`B ' 01 ? `B ( 01 ?f f `B ) 01 ? `B * 01 ? `B + 01 ? `B , 01 ? `B - 01 ? `B . 01 ? fB / 6o ?,P PfB 0 6o ?6fB 1 6o ? 6 fB 2 6o ?fB 3 6o ?  fB 4 6o ?fB 5 6o ?  fB 6 6o ?fB 7 6o ?  fB 8 6o ? fB 9 6o ?  fB : 6o ? fB ; 6o ? fB < 6o ? fB = 6o ? fB > 6o ? f fB ? 6o ? f fB @ 6o ?f L fB A 6o ? f L fB B 6o ?L 2fB C 6o ? L 2fB D 6o ?2fB E 6o ? 2 fB F 6o ?fB G 6o ?   H c $$ p0  <$@0    I 0   ,$0 n&The 2nd offer from the bank is $24000.,' 2  H  0޽h ? ̙33___PPT10{.i4I +1 (D' ` = @B D' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I%(+8+0+I0 +) 0 ((GL((  r  S § P   ' P  # #"F: P@   <ڧ  ?,  W100 000 @`  <p" ? ,  S 75n @`   <ȧ  ?,  R75 000 @`   <\  ?,  S 50n @`   <  ?,  V50 000 @`   <  ?, S 25n @`   <  ?,  V25 000 @`  <l  ?, c 10(n @`  <  ?,  V10 000 @`  <|!  ?, R 5n @`  <1  ?,6  Q5 000 @`  <-  ?6, d 1(nn @`  < 6o ? fB ? 6o ? fB @ 6o ? fB A 6o ? f fB B 6o ? f fB C 6o ?f L fB D 6o ? f L fB E 6o ?L 2fB F 6o ? L 2fB G 6o ?2fB H 6o ? 2 fB I 6o ?fB J 6o ?  H  0޽h ? ̙33___PPT10i.i4&+D=' ` = @B ++ .0 **HH**(  x  c $ P   ' P  # #"F: P@   <  ?,  W100 000 @`  <  ? ,  S 75n @`  <D  ?,  R75 000 @`  <4  ?,  S 50n @`  <  ?,  V50 000 @`   <\  ?, S 25n @`   <$  ?,  V25 000 @`   <  ?, c 10(n @`   <7  ?,  V10 000 @`   <9  ?, R 5n @`  <I  ?,6  Q5 000 @`  < L  ?6, d 1(nn @`  <x\  ?, f  S300 000 @`  <4_  ? ,f  d 200(n @`  < p  ?,  W200 000 @`  <tr  ? ,  d 100(n @`  <T  ?,L 2 W500 000 @`  <Ѕ  ?L ,2 X 400n @`  <ԕ  ?,f L  S400 000 @`  <h  ?f ,L  d 300(n @`  <T  ?,2  S750 000 @`  <죩  ?2, X 500n @`  <  ?,  Y 1 000 000   @`  <P  ?, X 750n @`  <Ω  ?,P 6 S1 000 @`  <ש  ?P,6 S.01  hh@`fB  6o ?P,P`B  01 ?6 6`B   01 ? fB ! 6o ?P6`B " 01 ?,P,fB # 6o ? P 6`B $ 01 ? `B % 01 ?L L `B & 01 ?2 2`B ' 01 ? `B ( 01 ?f f `B ) 01 ? `B * 01 ? `B + 01 ? `B , 01 ? `B - 01 ? `B . 01 ? fB / 6o ?,P PfB 0 6o ?6fB 1 6o ? 6 fB 2 6o ?fB 3 6o ?  fB 4 6o ?fB 5 6o ?  fB 6 6o ?fB 7 6o ?  fB 8 6o ? fB 9 6o ?  fB : 6o ? fB ; 6o ? fB < 6o ? fB = 6o ? fB > 6o ? f fB ? 6o ? f fB @ 6o ?f L fB A 6o ? f L fB B 6o ?L 2fB C 6o ? L 2fB D 6o ?2fB E 6o ? 2 fB F 6o ?fB G 6o ?   H 6L .` `  *Data from 2nd Deal: Mean= $112097 Median = $1000 Bank s offers: 1st Deal $11000 2nd Deal $24000Hd 8 H  0޽h ? ̙33___PPT10i.i4&+D=' ` = @B +2 /0 **IKZ*(  x  c $ P   ' P@  K #"F: P@   <  ?l @  W100 000 @`  <l  ? l  S 75n @`  <  ?l@  R75 000 @`  <  ?l  S 50n @`  <2  ?l@  V50 000 @`   <P5  ?l S 25n @`   <E  ?l@  V25 000 @`   <>  ?l c 10(n @`   <$X  ?l@  V10 000 @`   <Z  ?l R 5n @`  <j  ?l6@  Q5 000 @`  < 6o ? f fB ? 6o ?@ @ f fB @ 6o ?f L fB A 6o ?@ f @ L fB B 6o ?L 2fB C 6o ?@ L @ 2fB D 6o ?2fB E 6o ?@ 2@ fB F 6o ?fB G 6o ?@ @  I <t  p0  <$@0    J 0   ,$0 n&The 3rd offer from the bank is $50000.,' 2  H  0޽h ? ̙33___PPT10{.i4&+RGpD' ` = @B D' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*J%(+8+0+J0 +m 0 $$(  $r $ S P   r $ S P  H $ 0޽h ? ̙33y___PPT10Y+D=' ` = @B +* 00 ))pHL2)(  x  c $9 P   ' P@  # #"F: P@   <I  ?l @  W100 000 @`  <C  ? l  S 75n @`  <Z  ?l@  R75 000 @`  <]  ?l  S 50n @`  <m  ?l@  V50 000 @`   <f  ?l c 25(n @`   <2  ?l@  V25 000 @`   <|  ?l c 10(n @`   <  ?l@  V10 000 @`   <  ?l R 5n @`  <x  ?l6@  U5 000 @`  <4  ?6l d 1(nn @`  <  ?l @ f  W300 000 @`  <໬  ? lf  d 200(n @`  <̬  ?l @  W200 000 @`  <  ? l  d 100(n @`  <Ь  ?lL @ 2 W500 000 @`  <  ?L l2 X 400n @`  <<  ?lf @ L  S400 000 @`  <  ?f lL  d 300(n @`  <  ?l2@  S750 000 @`  <X  ?2l X 500n @`  <  ?l@  Y 1 000 000   @`  <$  ?l X 750n @`  <@+  ?lP@ 6 S1 000 @`  <4  ?Pl6 S.01  hh@`fB  6o ?PlP`B  01 ?6@ 6`B   01 ?@ fB ! 6o ?P6`B " 01 ?lPlfB # 6o ?@ P@ 6`B $ 01 ?@ `B % 01 ?L @ L `B & 01 ?2@ 2`B ' 01 ? @ `B ( 01 ?f @ f `B ) 01 ?@ `B * 01 ?@ `B + 01 ?@ `B , 01 ?@ `B - 01 ? @ `B . 01 ? @ fB / 6o ?lP@ PfB 0 6o ?6fB 1 6o ?@ 6@ fB 2 6o ?fB 3 6o ?@ @ fB 4 6o ?fB 5 6o ?@ @ fB 6 6o ?fB 7 6o ?@ @ fB 8 6o ? fB 9 6o ?@ @ fB : 6o ?  fB ; 6o ?@ @ fB < 6o ?  fB = 6o ?@ @ fB > 6o ? f fB ? 6o ?@ @ f fB @ 6o ?f L fB A 6o ?@ f @ L fB B 6o ?L 2fB C 6o ?@ L @ 2fB D 6o ?2fB E 6o ?@ 2@ fB F 6o ?fB G 6o ?@ @  L S F `    H  0޽h ? ̙33___PPT10i.i4&+D=' ` = @B +5 10 f+^+`IJ*(  x  c $0g P   ' P@  J #"F: P@   <  ?l @  W100 000 @`  <k  ? l  S 75n @`  <]  ?l@  R75 000 @`  <{  ?l  S 50n @`  <؉  ?l@  V50 000 @`   <X  ?l c 25(n @`   <  ?l@  V25 000 @`   <  ?l c 10(n @`   <,  ?l@  V10 000 @`   <  ?l R 5n @`  <  ?l6@  U5 000 @`  <@ĭ  ?6l d 1(nn @`  <,խ  ?l @ r  W300 000 @`  <׭  ? lr  d 200(n @`  <`  ?l @  W200 000 @`  <  ? l  d 100(n @`  <P  ?lX @ > W500 000 @`  <  ?X l> X 400n @`  <  ?lr @ X  S400 000 @`  <$  ?r lX  d 300(n @`  <"  ?l>@ $ S750 000 @`  <  ?>l$ X 500n @`  <h5  ?l$@  Y 1 000 000   @`  </  ?$l  X 750n @`  <XG  ?lP@ 6 S1 000 @`  <LP  ?Pl6 S.01  hh@`fB  6o ?PlP`B  01 ?6@ 6`B   01 ? @ fB ! 6o ?P6`B " 01 ?lPl fB # 6o ?@ P@ 6`B $ 01 ?$@ $`B % 01 ?X @ X `B & 01 ?>@ >`B ' 01 ? @ `B ( 01 ?r @ r `B ) 01 ?@ `B * 01 ?@ `B + 01 ?@ `B , 01 ?@ `B - 01 ? @ `B . 01 ? @ fB / 6o ?lP@ PfB 0 6o ?6fB 1 6o ?@ 6@ fB 2 6o ?fB 3 6o ?@ @ fB 4 6o ?fB 5 6o ?@ @ fB 6 6o ?fB 7 6o ?@ @ fB 8 6o ? fB 9 6o ?@ @ fB : 6o ?  fB ; 6o ?@ @ fB < 6o ?  fB = 6o ?@ @ fB > 6o ? r fB ? 6o ?@ @ r fB @ 6o ?r X fB A 6o ?@ r @ X fB B 6o ?X >fB C 6o ?@ X @ >fB D 6o ?>$fB E 6o ?@ >@ $fB F 6o ?$ fB G 6o ?@ $@  H <b  P P<$@0   x I 0he P ,$0 What percentage of the mean value (12%, 21%, 36 % previously in order) do you think the offer will be? The 4th offer from the bank is $62000.:g0Z( 2l !H  0޽h ? ̙33  ___PPT10 .i4&+1 (DS ' ` = @B D ' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Ig%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ig%(+8+0+I0 +* :0 ))PHJ@2)(  @x @ c $D P   ' P@  @# #"F: P@  @ <  ?l @  W100 000 @` @ <  ? l  S 75n @` @ <  ?l@  R75 000 @` @ <  ?l  S 50n @` @ <  ?l@  V50 000 @`  @ <@  ?l c 25(n @`  @ <,Ǯ  ?l@  V25 000 @`  @ <ɮ  ?l c 10(n @`  @ <`ڮ  ?l@  V10 000 @`  @ <ܮ  ?l R 5n @` @ <  ?l6@  U5 000 @` @ <t  ?6l d 1(nn @` @ <  ?l @ r  W300 000 @` @ <|  ? lr  d 200(n @` @ <  ?l @  W200 000 @` @ <<  ? l  d 100(n @` @ <  ?lX @ > W500 000 @` @ <0"  ?X l> X 400n @` @ <:  ?lr @ X  S400 000 @` @ <<  ?r lX  d 300(n @` @ <M  ?l>@ $ S750 000 @` @ <(H  ?>l$ X 500n @` @ <_  ?l$@  Y 1 000 000   @` @ <b  ?$l  X 750n @` @ <r  ?lP@ 6 S1 000 @` @ <{  ?Pl6 S.01  hh@`fB @ 6o ?PlP`B @ 01 ?6@ 6`B  @ 01 ? @ fB !@ 6o ?P6`B "@ 01 ?lPl fB #@ 6o ?@ P@ 6`B $@ 01 ?$@ $`B %@ 01 ?X @ X `B &@ 01 ?>@ >`B '@ 01 ? @ `B (@ 01 ?r @ r `B )@ 01 ?@ `B *@ 01 ?@ `B +@ 01 ?@ `B ,@ 01 ?@ `B -@ 01 ? @ `B .@ 01 ? @ fB /@ 6o ?lP@ PfB 0@ 6o ?6fB 1@ 6o ?@ 6@ fB 2@ 6o ?fB 3@ 6o ?@ @ fB 4@ 6o ?fB 5@ 6o ?@ @ fB 6@ 6o ?fB 7@ 6o ?@ @ fB 8@ 6o ? fB 9@ 6o ?@ @ fB :@ 6o ?  fB ;@ 6o ?@ @ fB <@ 6o ?  fB =@ 6o ?@ @ fB >@ 6o ? r fB ?@ 6o ?@ @ r fB @@ 6o ?r X fB A@ 6o ?@ r @ X fB B@ 6o ?X >fB C@ 6o ?@ X @ >fB D@ 6o ?>$fB E@ 6o ?@ >@ $fB F@ 6o ?$ fB G@ 6o ?@ $@  J@ S  0    H @ 0޽h ? ̙33___PPT10i.i4&+D=' ` = @B + * 20 "))@GJ((  x  c $ P   ' P@  # #"F: P@   <H  ?l @  W100 000 @`  <$  ? l  S 75n @`  <  ?l@  V75 000 @`  <`İ  ?l  S 50n @`  <԰  ?l@  V50 000 @`   <Ͱ  ?l c 25(n @`   <  ?l@  V25 000 @`   <  ?l c 10(n @`   <  ?l@  V10 000 @`   <  ?l R 5n @`  <  ?l6@  U5 000 @`  <  ?6l d 1(nn @`  <"  ?l @ r  W300 000 @`  <$  ? lr  d 200(n @`  <5  ?l @  W200 000 @`  <H8  ? l  d 100(n @`  <I  ?lX @ > W500 000 @`  <C  ?X l> X 400n @`  <T[  ?lr @ X  S400 000 @`  <]  ?r lX  d 300(n @`  <n  ?l>@ $ W750 000 @`  <,q  ?>l$ X 500n @`  <D  ?l$@  Y 1 000 000   @`  <D  ?$l  X 750n @`  <<  ?lP@ 6 S1 000 @`  <X  ?Pl6 S.01  hh@`fB  6o ?PlP`B  01 ?6@ 6`B   01 ? @ fB ! 6o ?P6`B " 01 ?lPl fB # 6o ?@ P@ 6`B $ 01 ?$@ $`B % 01 ?X @ X `B & 01 ?>@ >`B ' 01 ? @ `B ( 01 ?r @ r `B ) 01 ?@ `B * 01 ?@ `B + 01 ?@ `B , 01 ?@ `B - 01 ? @ `B . 01 ? @ fB / 6o ?lP@ PfB 0 6o ?6fB 1 6o ?@ 6@ fB 2 6o ?fB 3 6o ?@ @ fB 4 6o ?fB 5 6o ?@ @ fB 6 6o ?fB 7 6o ?@ @ fB 8 6o ? fB 9 6o ?@ @ fB : 6o ?  fB ; 6o ?@ @ fB < 6o ?  fB = 6o ?@ @ fB > 6o ? r fB ? 6o ?@ @ r fB @ 6o ?r X fB A 6o ?@ r @ X fB B 6o ?X >fB C 6o ?@ X @ >fB D 6o ?>$fB E 6o ?@ >@ $fB F 6o ?$ fB G 6o ?@ $@ H  0޽h ? ̙33___PPT10i.i4&+D=' ` = @B + 0 0 4(   r   S `x P     0 6A ?0   H   0޽h ? ̙33___PPT10i.o4@+D=' ` = @B +2 30 ** II*(  x  c $̱ P   ' P@  # #"F: P@   <ر  ?l @  W100 000 @`  <ұ  ? l  S 75n @`  <  ?l@  V75 000 @`  <$  ?l  S 50n @`  <\  ?l@  V50 000 @`   <  ?l c 25(n @`   <  ?l@  V25 000 @`   <\  ?l c 10(n @`   <  ?l@  V10 000 @`   <@'  ?l R 5n @`  <6  ?l6@  U5 000 @`  <h9  ?6l d 1(nn @`  <TJ  ?l @ r  W300 000 @`  <L  ? lr  d 200(n @`  <]  ?l @  W200 000 @`  <P`  ? l  d 100(n @`  <tp  ?lX @ > W500 000 @`  <Xs  ?X l> X 400n @`  <܃  ?lr @ X  S400 000 @`  <|  ?r lX  d 300(n @`  <x  ?l>@ $ W750 000 @`  <@  ?>l$ X 500n @`  <  ?l$@  Y 1 000 000   @`  <  ?$l  X 750n @`  <  ?lP@ 6 S1 000 @`  <tĴ  ?Pl6 S.01  hh@`fB  6o ?PlP`B  01 ?6@ 6`B   01 ? @ fB ! 6o ?P6`B " 01 ?lPl fB # 6o ?@ P@ 6`B $ 01 ?$@ $`B % 01 ?X @ X `B & 01 ?>@ >`B ' 01 ? @ `B ( 01 ?r @ r `B ) 01 ?@ `B * 01 ?@ `B + 01 ?@ `B , 01 ?@ `B - 01 ? @ `B . 01 ? @ fB / 6o ?lP@ PfB 0 6o ?6fB 1 6o ?@ 6@ fB 2 6o ?fB 3 6o ?@ @ fB 4 6o ?fB 5 6o ?@ @ fB 6 6o ?fB 7 6o ?@ @ fB 8 6o ? fB 9 6o ?@ @ fB : 6o ?  fB ; 6o ?@ @ fB < 6o ?  fB = 6o ?@ @ fB > 6o ? r fB ? 6o ?@ @ r fB @ 6o ?r X fB A 6o ?@ r @ X fB B 6o ?X >fB C 6o ?@ X @ >fB D 6o ?>$fB E 6o ?@ >@ $fB F 6o ?$ fB G 6o ?@ $@  H <ִ   <$@0    I 0ٴ L  ,$D0 o'The 5th offer from the bank is $33000.,( 2 !H  0޽h ? ̙33^V___PPT106.i4&+ID ' ` = @B D' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I(%(+* 40 &)) GJ ((   x   c $\ P   ' P@   # #"F: P@    <  ?l @  W100 000 @`   <  ? l  S 75n @`   <H  ?l@  V75 000 @`   <  ?l  S 50n @`   <  ?l@  V50 000 @`   <L!  ?l c 25(n @`   <+  ?l@  V25 000 @`   <@;  ?l c 10(n @`   <,L  ?l@  V10 000 @`   <F  ?l R 5n @`   <d^  ?l6@  U5 000 @`   <`  ?6l d 1(nn @`   <q  ?l @ r  W300 000 @`   < W500 000 @`   <  ?X l> X 400n @`   <̪  ?lr @ X  S400 000 @`   <ģ  ?r lX  d 300(n @`   <h  ?l>@ $ W750 000 @`   <0  ?>l$ X 500n @`   <ж  ?l$@  Y 1 000 000   @`   <ʶ  ?$l  X 750n @`   <p  ?lP@ 6 S1 000 @`   <d  ?Pl6 W.01  hh@`fB   6o ?PlP`B   01 ?6@ 6`B   01 ? @ fB !  6o ?P6`B "  01 ?lPl fB #  6o ?@ P@ 6`B $  01 ?$@ $`B %  01 ?X @ X `B &  01 ?>@ >`B '  01 ? @ `B (  01 ?r @ r `B )  01 ?@ `B *  01 ?@ `B +  01 ?@ `B ,  01 ?@ `B -  01 ? @ `B .  01 ? @ fB /  6o ?lP@ PfB 0  6o ?6fB 1  6o ?@ 6@ fB 2  6o ?fB 3  6o ?@ @ fB 4  6o ?fB 5  6o ?@ @ fB 6  6o ?fB 7  6o ?@ @ fB 8  6o ? fB 9  6o ?@ @ fB :  6o ?  fB ;  6o ?@ @ fB <  6o ?  fB =  6o ?@ @ fB >  6o ? r fB ?  6o ?@ @ r fB @  6o ?r X fB A  6o ?@ r @ X fB B  6o ?X >fB C  6o ?@ X @ >fB D  6o ?>$fB E  6o ?@ >@ $fB F  6o ?$ fB G  6o ?@ $@ H   0޽h ? ̙33___PPT10i.i4&+D=' ` = @B +2 50 +*II*(  x  c $ P   ' P@  # #"F: P@   <  ?l @  W100 000 @`  <  ? l  S 75n @`  <H(  ?l@  V75 000 @`  <+  ?l  S 50n @`  <T8  ?l@  V50 000 @`   <2  ?l c 25(n @`   <`;  ?l@  V25 000 @`   <N  ?l c 10(n @`   <p^  ?l@  V10 000 @`   <`  ?l R 5n @`  <p  ?l6@  U5 000 @`  <s  ?6l d 1(nn @`  <p  ?l @ r  W300 000 @`  <Ć  ? lr  d 200(n @`  <|  ?l @  W200 000 @`  <  ? l  d 100(n @`  <l  ?lX @ > W500 000 @`  <l  ?X l> X 400n @`  <d  ?lr @ X  S400 000 @`  <Է  ?r lX  d 300(n @`  <з  ?l>@ $ W750 000 @`  <L˷  ?>l$ X 500n @`  <8  ?l$@  Y 1 000 000   @`  <hݷ  ?$l  X 750n @`  <(  ?lP@ 6 S1 000 @`  <  ?Pl6 W.01  hh@`fB  6o ?PlP`B  01 ?6@ 6`B   01 ? @ fB ! 6o ?P6`B " 01 ?lPl fB # 6o ?@ P@ 6`B $ 01 ?$@ $`B % 01 ?X @ X `B & 01 ?>@ >`B ' 01 ? @ `B ( 01 ?r @ r `B ) 01 ?@ `B * 01 ?@ `B + 01 ?@ `B , 01 ?@ `B - 01 ? @ `B . 01 ? @ fB / 6o ?lP@ PfB 0 6o ?6fB 1 6o ?@ 6@ fB 2 6o ?fB 3 6o ?@ @ fB 4 6o ?fB 5 6o ?@ @ fB 6 6o ?fB 7 6o ?@ @ fB 8 6o ? fB 9 6o ?@ @ fB : 6o ?  fB ; 6o ?@ @ fB < 6o ?  fB = 6o ?@ @ fB > 6o ? r fB ? 6o ?@ @ r fB @ 6o ?r X fB A 6o ?@ r @ X fB B 6o ?X >fB C 6o ?@ X @ >fB D 6o ?>$fB E 6o ?@ >@ $fB F 6o ?$ fB G 6o ?@ $@  H <    <$@0    I 0 L  ,$D0 o'The 6th offer from the bank is $68000.,( 2 !H  0޽h ? ̙33^V___PPT106.i4&+ID ' ` = @B D' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I(%(+* 60 *)")GJ((  x  c $1 P   ( P@  # #"F: P@   <<  ?l @  W100 000 @`  <6  ? l  S 75n @`  <M  ?l@  V75 000 @`  <P  ?l  S 50n @`  <TZ  ?l@  V50 000 @`   <b  ?l c 25(n @`   <s  ?l@  V25 000 @`   <,v  ?l c 10(n @`   <  ?l@  V10 000 @`   <  ?l R 5n @`  <  ?l6@  U5 000 @`  <  ?6l d 1(nn @`  <,  ?l @ r  W300 000 @`  <  ? lr  d 200(n @`  <谸  ?l @  W200 000 @`  <¸  ? l  d 100(n @`  <Ӹ  ?lX @ > W500 000 @`  <Tθ  ?X l> X 400n @`  <  ?lr @ X  S400 000 @`  <H  ?r lX  d 300(n @`  <  ?l>@ $ W750 000 @`  <  ?>l$ X 500n @`  <  ?l$@  Y 1 000 000   @`  <  ?$l  X 750n @`  <  ?lP@ 6 W1 000 @`  <(  ?Pl6 W.01  hh@`fB  6o ?PlP`B  01 ?6@ 6`B   01 ? @ fB ! 6o ?P6`B " 01 ?lPl fB # 6o ?@ P@ 6`B $ 01 ?$@ $`B % 01 ?X @ X `B & 01 ?>@ >`B ' 01 ? @ `B ( 01 ?r @ r `B ) 01 ?@ `B * 01 ?@ `B + 01 ?@ `B , 01 ?@ `B - 01 ? @ `B . 01 ? @ fB / 6o ?lP@ PfB 0 6o ?6fB 1 6o ?@ 6@ fB 2 6o ?fB 3 6o ?@ @ fB 4 6o ?fB 5 6o ?@ @ fB 6 6o ?fB 7 6o ?@ @ fB 8 6o ? fB 9 6o ?@ @ fB : 6o ?  fB ; 6o ?@ @ fB < 6o ?  fB = 6o ?@ @ fB > 6o ? r fB ? 6o ?@ @ r fB @ 6o ?r X fB A 6o ?@ r @ X fB B 6o ?X >fB C 6o ?@ X @ >fB D 6o ?>$fB E 6o ?@ >@ $fB F 6o ?$ fB G 6o ?@ $@ H  0޽h ? ̙33___PPT10i.i4&+D=' ` = @B +2 70 +*II$*(  $x $ c $XG P   ( P@  $# #"F: P@  $ <a ?l @  W100 000 @` $ <O  ? l  S 75n @` $ <Z  ?l@  V75 000 @` $ <Xb  ?l  S 50n @` $ <r  ?l@  V50 000 @`  $ <Xt  ?l c 25(n @`  $ <  ?l@  V25 000 @`  $ <ć  ?l c 10(n @`  $ <  ?l@  V10 000 @`  $ <(  ?l R 5n @` $ <`  ?l6@  U5 000 @` $ <X  ?6l d 1(nn @` $ <  ?l @ r  W300 000 @` $ <ܿ  ? lr  d 200(n @` $ <tй  ?l @  W200 000 @` $ <Xӹ  ? l  d 100(n @` $ <  ?lX @ > W500 000 @` $ <<  ?X l> X 400n @` $ <,  ?lr @ X  S400 000 @` $ <(  ?r lX  d 300(n @` $ <  ?l>@ $ W750 000 @` $ <p  ?>l$ X 500n @` $ <,  ?l$@  Y 1 000 000   @` $ <  ?$l  X 750n @` $ <l0  ?lP@ 6 W1 000 @` $ <9  ?Pl6 W.01  hh@`fB $ 6o ?PlP`B $ 01 ?6@ 6`B  $ 01 ? @ fB !$ 6o ?P6`B "$ 01 ?lPl fB #$ 6o ?@ P@ 6`B $$ 01 ?$@ $`B %$ 01 ?X @ X `B &$ 01 ?>@ >`B '$ 01 ? @ `B ($ 01 ?r @ r `B )$ 01 ?@ `B *$ 01 ?@ `B +$ 01 ?@ `B ,$ 01 ?@ `B -$ 01 ? @ `B .$ 01 ? @ fB /$ 6o ?lP@ PfB 0$ 6o ?6fB 1$ 6o ?@ 6@ fB 2$ 6o ?fB 3$ 6o ?@ @ fB 4$ 6o ?fB 5$ 6o ?@ @ fB 6$ 6o ?fB 7$ 6o ?@ @ fB 8$ 6o ? fB 9$ 6o ?@ @ fB :$ 6o ?  fB ;$ 6o ?@ @ fB <$ 6o ?  fB =$ 6o ?@ @ fB >$ 6o ? r fB ?$ 6o ?@ @ r fB @$ 6o ?r X fB A$ 6o ?@ r @ X fB B$ 6o ?X >fB C$ 6o ?@ X @ >fB D$ 6o ?>$fB E$ 6o ?@ >@ $fB F$ 6o ?$ fB G$ 6o ?@ $@  H$ <$L   <$@0    I$ 0N L  ,$D0 o'The 7th offer from the bank is $90000.,( 2 !H $ 0޽h ? ̙33^V___PPT106.i4&+ID ' ` = @B D' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H$%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I$(%(+ 0  <^(  <r < S 0r P     <0  `A  ? ?   H < 0޽h ? ̙33___PPT10i.t4+D=' ` = @B +#* 80 :)2)GJ0((  0x 0 c $x P   ( P@  0# #"F: P@  0 <m  ?l @  W100 000 @` 0 <P  ? l  S 75n @` 0 <  ?l@  V75 000 @` 0 <  ?l  c 50(n @` 0 <  ?l@  V50 000 @`  0 <  ?l c 25(n @`  0 <tȻ  ?l@  V25 000 @`  0 <ʻ  ?l c 10(n @`  0 <ۻ  ?l@  V10 000 @`  0 <$޻  ?l R 5n @` 0 <(  ?l6@  U5 000 @` 0 <  ?6l d 1(nn @` 0 <  ?l @ r  W300 000 @` 0 <  ? lr  d 200(n @` 0 <D  ?l @  W200 000 @` 0 <  ? l  d 100(n @` 0 <(  ?lX @ > W500 000 @` 0 <D+  ?X l> X 400n @` 0 <|;  ?lr @ X  S400 000 @` 0 <t4  ?r lX  d 300(n @` 0 <N  ?l>@ $ W750 000 @` 0 <P  ?>l$ X 500n @` 0 <`  ?l$@  Y 1 000 000   @` 0 <[  ?$l  X 750n @` 0 <r  ?lP@ 6 W1 000 @` 0 <{  ?Pl6 W.01  hh@`fB 0 6o ?PlP`B 0 01 ?6@ 6`B  0 01 ? @ fB !0 6o ?P6`B "0 01 ?lPl fB #0 6o ?@ P@ 6`B $0 01 ?$@ $`B %0 01 ?X @ X `B &0 01 ?>@ >`B '0 01 ? @ `B (0 01 ?r @ r `B )0 01 ?@ `B *0 01 ?@ `B +0 01 ?@ `B ,0 01 ?@ `B -0 01 ? @ `B .0 01 ? @ fB /0 6o ?lP@ PfB 00 6o ?6fB 10 6o ?@ 6@ fB 20 6o ?fB 30 6o ?@ @ fB 40 6o ?fB 50 6o ?@ @ fB 60 6o ?fB 70 6o ?@ @ fB 80 6o ? fB 90 6o ?@ @ fB :0 6o ?  fB ;0 6o ?@ @ fB <0 6o ?  fB =0 6o ?@ @ fB >0 6o ? r fB ?0 6o ?@ @ r fB @0 6o ?r X fB A0 6o ?@ r @ X fB B0 6o ?X >fB C0 6o ?@ X @ >fB D0 6o ?>$fB E0 6o ?@ >@ $fB F0 6o ?$ fB G0 6o ?@ $@ H 0 0޽h ? ̙33___PPT10i.i4&+D=' ` = @B +2 90 ++II8*(  8x 8 c $Ȝ P   ( P@  8# #"F: P@  8 <Ȫ  ?l @  W100 000 @` 8 <ȭ  ? l  S 75n @` 8 <  ?l@  V75 000 @` 8 <0  ?l  c 50(n @` 8 <Tν  ?l@  V50 000 @`  8 <н  ?l c 25(n @`  8 <  ?l@  V25 000 @`  8 <  ?l c 10(n @`  8 <  ?l@  V10 000 @`  8 <D  ?l R 5n @` 8 <  ?l6@  U5 000 @` 8 <  ?6l d 1(nn @` 8 <  ?l @ r  W300 000 @` 8 <H  ? lr  d 200(n @` 8 <.  ?l @  W200 000 @` 8 <41  ? l  d 100(n @` 8 <A  ?lX @ > W500 000 @` 8 <D  ?X l> X 400n @` 8 <T  ?lr @ X  S400 000 @` 8 <O  ?r lX  d 300(n @` 8 <g  ?l>@ $ W750 000 @` 8 <Pj  ?>l$ X 500n @` 8 <z  ?l$@  Y 1 000 000   @` 8 <u  ?$l  X 750n @` 8 <  ?lP@ 6 W1 000 @` 8 <  ?Pl6 W.01  hh@`fB 8 6o ?PlP`B 8 01 ?6@ 6`B  8 01 ? @ fB !8 6o ?P6`B "8 01 ?lPl fB #8 6o ?@ P@ 6`B $8 01 ?$@ $`B %8 01 ?X @ X `B &8 01 ?>@ >`B '8 01 ? @ `B (8 01 ?r @ r `B )8 01 ?@ `B *8 01 ?@ `B +8 01 ?@ `B ,8 01 ?@ `B -8 01 ? @ `B .8 01 ? @ fB /8 6o ?lP@ PfB 08 6o ?6fB 18 6o ?@ 6@ fB 28 6o ?fB 38 6o ?@ @ fB 48 6o ?fB 58 6o ?@ @ fB 68 6o ?fB 78 6o ?@ @ fB 88 6o ? fB 98 6o ?@ @ fB :8 6o ?  fB ;8 6o ?@ @ fB <8 6o ?  fB =8 6o ?@ @ fB >8 6o ? r fB ?8 6o ?@ @ r fB @8 6o ?r X fB A8 6o ?@ r @ X fB B8 6o ?X >fB C8 6o ?@ X @ >fB D8 6o ?>$fB E8 6o ?@ >@ $fB F8 6o ?$ fB G8 6o ?@ $@  H8 <h   <$@0    I8 0 L  ,$D0 p(The 8th offer from the bank is $137000.,) 2 "H 8 0޽h ? ̙33^V___PPT106.i4&+ID ' ` = @B D' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H8%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H8%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I8)%(+s B0 @*(  r  S  P   x  <A  ??t  H  0޽h ? ̙33y___PPT10Y+D=' ` = @B +} !0 D$(  Dr D S Ҿ P   r D S (վ    H D 0޽h ? ̙33___PPT10i.u4XjJ+D=' ` = @B +/ ;0 ))HIHT)(  Hx H c $d P   ' P@  H# #"F: P@  H <  ?l @  W100 000 @` H <D  ? l  W 75n @` H <  ?l@  V75 000 @` H <  ?l  W 50n @` H <  ?l@  V50 000 @`  H <8  ?l W 25n @`  H <p-  ?l@  V25 000 @`  H <h&  ?l W 10n @`  H <|?  ?l@  V10 000 @`  H <DB  ?l V 5n @` H <hR  ?l6@  U5 000 @` H <L  ?6l X 1nn @` H <d  ?l @ r  W300 000 @` H < g  ? lr  X 200n @` H <Xw  ?l @  W200 000 @` H <Pp  ? l  X 100n @` H <d  ?lX @ > W500 000 @` H <,  ?X l> X 400n @` H <P  ?lr @ X  W400 000 @` H <  ?r lX  X 300n @` H <t  ?l>@ $ W750 000 @` H <  ?>l$ X 500n @` H <H  ?l$@  Y 2 000 000   @` H <  ?$l  X 750n @` H <l  ?lP@ 6 W1 000 @` H <t  ?Pl6 W.01  hh@`fB H 6o ?PlP`B H 01 ?6@ 6`B  H 01 ? @ fB !H 6o ?P6`B "H 01 ?lPl fB #H 6o ?@ P@ 6`B $H 01 ?$@ $`B %H 01 ?X @ X `B &H 01 ?>@ >`B 'H 01 ? @ `B (H 01 ?r @ r `B )H 01 ?@ `B *H 01 ?@ `B +H 01 ?@ `B ,H 01 ?@ `B -H 01 ? @ `B .H 01 ? @ fB /H 6o ?lP@ PfB 0H 6o ?6fB 1H 6o ?@ 6@ fB 2H 6o ?fB 3H 6o ?@ @ fB 4H 6o ?fB 5H 6o ?@ @ fB 6H 6o ?fB 7H 6o ?@ @ fB 8H 6o ? fB 9H 6o ?@ @ fB :H 6o ?  fB ;H 6o ?@ @ fB H 6o ? r fB ?H 6o ?@ @ r fB @H 6o ?r X fB AH 6o ?@ r @ X fB BH 6o ?X >fB CH 6o ?@ X @ >fB DH 6o ?>$fB EH 6o ?@ >@ $fB FH 6o ?$ fB GH 6o ?@ $@  HH <   0 P<$D0   H H 0޽h ? ̙33___PPT10.i4&+#,D~' ` = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*HH_%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*HH`%(+6 <0 ))HIPb)(  Px P c $$ P   ' P@  P# #"F: P@  P <  ?l @  W100 000 @` P <  ? l  W 75n @` P <)  ?l@  V75 000 @` P <X"  ?l  W 50n @` P <<  ?l@  V50 000 @`  P <p>  ?l W 25n @`  P <N  ?l@  V25 000 @`  P <G  ?l W 10n @`  P <`  ?l@  V10 000 @`  P <|c  ?l V 5n @` P <s  ?l6@  U5 000 @` P <n  ?6l X 1nn @` P <ą  ?l @ r  W300 000 @` P <X  ? lr  X 200n @` P <  ?l @  W200 000 @` P <  ? l  X 100n @` P <  ?lX @ > W500 000 @` P <d  ?X l> X 400n @` P <  ?lr @ X  W400 000 @` P <  ?r lX  X 300n @` P <  ?l>@ $ W750 000 @` P <(  ?>l$ X 500n @` P <  ?l$@  Y 2 000 000   @` P <  ?$l  X 750n @` P <  ?lP@ 6 W1 000 @` P <  ?Pl6 W.01  hh@`fB P 6o ?PlP`B P 01 ?6@ 6`B  P 01 ? @ fB !P 6o ?P6`B "P 01 ?lPl fB #P 6o ?@ P@ 6`B $P 01 ?$@ $`B %P 01 ?X @ X `B &P 01 ?>@ >`B 'P 01 ? @ `B (P 01 ?r @ r `B )P 01 ?@ `B *P 01 ?@ `B +P 01 ?@ `B ,P 01 ?@ `B -P 01 ? @ `B .P 01 ? @ fB /P 6o ?lP@ PfB 0P 6o ?6fB 1P 6o ?@ 6@ fB 2P 6o ?fB 3P 6o ?@ @ fB 4P 6o ?fB 5P 6o ?@ @ fB 6P 6o ?fB 7P 6o ?@ @ fB 8P 6o ? fB 9P 6o ?@ @ fB :P 6o ?  fB ;P 6o ?@ @ fB P 6o ? r fB ?P 6o ?@ @ r fB @P 6o ?r X fB AP 6o ?@ r @ X fB BP 6o ?X >fB CP 6o ?@ X @ >fB DP 6o ?>$fB EP 6o ?@ >@ $fB FP 6o ?$ fB GP 6o ?@ $@  HP <   <$D0   (DH P 0޽h ? ̙33 ~ ___PPT10^ .i4&+#,D2 ' ` = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*HP.%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*HP.[%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*HP\p%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*HPp%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*HP%(+q* =0 ))HKX)(  Xx X c $@ P   ' P@  KX #"F: P@  X <N  ?l @  S200 000 @` X <hI  ? l  W 75n @` X <`^  ?l@  V75 000 @` X <`  ?l  W 50n @` X <,q  ?l@  V50 000 @`  X <$j  ?l W 25n @`  X <8  ?l@  V25 000 @`  X <  ?l W 10n @`  X <$  ?l@  V10 000 @`  X <  ?l V 5n @` X <H  ?l6@  U5 000 @` X <ܪ  ?6l X 1nn @` X <  ?l @ r  S600 000 @` X <  ? lr  X 200n @` X <  ?l @  S400 000 @` X <  ? l  X 100n @` X <|  ?lX @ > U 1 000 000   @` X <  ?X l> X 400n @` X <  ?lr @ X  S800 000 @` X <,  ?r lX  X 300n @` X <  ?l>@ $ U 1 500 000   @` X <  ?>l$ X 500n @` X <  ?l$@  U 2 000 000   @` X <l  ?$l  X 750n @` X <  ?lP@ 6 W1 000 @` X <  ?Pl6 W.01  hh@`fB X 6o ?PlP`B X 01 ?6@ 6`B  X 01 ? @ fB !X 6o ?P6`B "X 01 ?lPl fB #X 6o ?@ P@ 6`B $X 01 ?$@ $`B %X 01 ?X @ X `B &X 01 ?>@ >`B 'X 01 ? @ `B (X 01 ?r @ r `B )X 01 ?@ `B *X 01 ?@ `B +X 01 ?@ `B ,X 01 ?@ `B -X 01 ? @ `B .X 01 ? @ fB /X 6o ?lP@ PfB 0X 6o ?6fB 1X 6o ?@ 6@ fB 2X 6o ?fB 3X 6o ?@ @ fB 4X 6o ?fB 5X 6o ?@ @ fB 6X 6o ?fB 7X 6o ?@ @ fB 8X 6o ? fB 9X 6o ?@ @ fB :X 6o ?  fB ;X 6o ?@ @ fB X 6o ? r fB ?X 6o ?@ @ r fB @X 6o ?r X fB AX 6o ?@ r @ X fB BX 6o ?X >fB CX 6o ?@ X @ >fB DX 6o ?>$fB EX 6o ?@ >@ $fB FX 6o ?$ fB GX 6o ?@ $@  HX <O      (HH X 0޽h ? ̙33___PPT10i.i4&+D=' ` = @B +6 @0 **pHHp)(  px p c $| P   ( P@  p# #"F: P@  p <   ?"`l @  S200 000 @` p <j  ? l  W 75n @` p <x  ?l@  V75 000 @` p <Ts  ?l  W 50n @` p <  ?l@  V50 000 @`  p <x  ?l W 25n @`  p <  ?l@  V25 000 @`  p <L  ?l W 10n @`  p <  ?l@  V10 000 @`  p <|  ?l V 5n @` p <  ?l6@  U5 000 @` p <X  ?6l X 1nn @` p <|   ?"`l @ r  S600 000 @` p <  ? lr  X 200n @` p <   ?"`l @  S400 000 @` p <4  ? l  X 100n @` p <t   ?"`lX @ > U 1 000 000   @` p <  ?X l> X 400n @` p <   ?"`lr @ X  S800 000 @` p <l  ?r lX  X 300n @` p <   ?"`l>@ $ U 1 500 000   @` p <  ?>l$ X 500n @` p </   ?"`l$@  U 2 000 000   @` p <t2  ?$l  X 750n @` p <B  ?lP@ 6 W1 000 @` p <;  ?Pl6 W.01  hh@`fB p 6o ?PlP`B p 01 ?6@ 6`B  p 01 ? 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̙33 ~ ___PPT10^ .i4&+#,D2 ' ` = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Hx%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Hx.%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Hx.Z%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Hx[o%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Hxo%(+ 0 @H(  HX H C Dv    H S D     H H 02m.F ? 3380___PPT10.d4rq0 4,@L(  LX L C Dv   , L S HD    `Extremely popular game show on NBC hosted by Howie Mandel A contestant chooses 1 of 26 numbered cases and keeps that case (dollar amounts range from 1 cent to 1 million dollars.) The contestant then opens a certain number of cases he did not open to see what dollar amounts he didn t win After seeing these dollar amounts that he did not win, he may either open some more cases or take an offer from  the banker to buy back his case. If this offer is accepted, the contestant sells back his case and takes the offered amount of money and the game is over. (0Z0 -H L 02m.F ? 3380___PPT10.d4q0 d.(  dX d C Dv    d S T6 D    0Why are the deals getting  better? Player performance (he s knocking out more small numbers than larger ones) NBC wants players to go through to the end for higher suspense/ratings and hence they don t want everyone taking these early deals NBC is trying to minimize their risk of losing a  big value perhaps any value over a certain threshold ($300,000 and above for example.) Any other ideas? H d 02m.F ? 3380___PPT10.d4q$0 P4(  X  C Dv     S D    6Enter the 26 dollar amounts into your calculator and compute the mean and the median dollar amounts to win. On the TI-84, click on the STAT button (2nd row from the top in the middle). EDIT should be highlighted with 1:Edit highlighted. Click ENTER. In List L1, enter the values. Type in .01 ENTER in L1(1), 1 ENTER in L1(2), & 1000000 ENTER in L1(26). LlZZl)H  02m.F ? 3380___PPT10.g4̿0 LD`(  X  C Dv   D  S D    Once all 26 values are entered into the calculator, please check that the values are correct. One method to validate is to check the stats. Click on STAT and move the curser to the right  > and CALC is highlighted. Click ENTER and  1-Var Stats will be displayed . Click ENTER again and the stats will be displayed. >ZZ  H  02m.F ? 3380___PPT10.g4̿0 -%p(  X  C Dv   %  S JD    We now need to take $10000 off of our game board. Go to STAT and EDIT click ENTER. Scroll down to L1(16)=10000 and click DEL. This will delete this value from our list. H  02m.F ? 3380___PPT10.g4̿0 UM(  X  C Dv   M  S C D    After the first 6 cases are opened, click again on STAT, go > to highlight CALC and click ENTER on 1-Var Stats and click enter again to find the stats on the remaining values  H  02m.F ? 3380___PPT10.h4780 H(  X  C Dv     S 0 D    JWhat do you think will happen to the offer from the bank? How many values that he eliminated were above the mean value of around $95000 after the first offer and how many were below the mean value. What do you think will happen to the mean of his remaining values?&::H  02m.F ? 3380___PPT10.i4ʸ 0 (  X  C Dv     S 0 D     H  02m.F ? 3380___PPT10.i4@%0 0(` (   X   C Dv   (   S Ʊ D    Now the mathematics should be visible and intuitive with each case removed. Previous percentages of means have been 12, 21, 36, 41% of the mean& do you think this pattern will continue now that he had a bad round? H   02m.F ? 3380___PPT10.p40v 0 T(  TX T C Dv    T S  D    6The offer jumped to $90000, nearly 90% of the mean value of the remaining amounts, up from $68000, nearly 85% of the mean value of the remaining amounts before. Even if he eliminates the smallest amount and receives an offer of 100% of the mean, how much will the next offer be? Is the risk worth the reward? 6H T 02m.F ? 3380___PPT10.v4 F]"0 <4`(  `X ` C Dv   4 ` S ߾ D    RWhat do you think of the contestant s decisions? Intuitively? Mathematically? Did he follow the correct mathematical steps? Do you think you could follow along mathematically and make logical conclusions despite the pressure of the cameras and lack of calculators/computers? Question: Does it really matter what dollar amount was in the case that the contestant chose at the beginning of the game? Is contestant A, who chose a case with $100000 in it guaranteed to win more money on this show than a contestant B who chose a case with $300 in it. `PPH ` 02m.F ? 3380___PPT10.v4 F]r.0 (  ^  S Dv     c $pv D    xLet s get a show of hands as to who: Would sell back their case for $11000 Would say  no deal Don t forget that once you sell your case, the game is over and if you don t sell your case, you may not get an offer as high as this one in the next bank offer Z%:%:   H  02m.F ? 3380___PPT10.h47/0 (  ^  S Dv     c $ǧ D    dWhich appears to fluctuate more, the mean or the median? Last time the bank offered $11,000 when the mean was $95405 and the median was $875. What do you think the offer will be now? Would you sell back the case at this stage of the game& if you already sold it before for $11000, you re out of the game J $30H  02m.F ? 3380___PPT10.i4ʸ00 vn(  ^  S Dv   h  c $ D    Looking at the game board, what do you notice as a pattern& perhaps some gaps developing. How many values did he remove in this round above last round s mean of $112000. How many were above? What do you think will happen to the banker s offer? H  02m.F ? 3380___PPT10.i4@%,10 <(  ^  S Dv     c $( D    2Looking at the game board, what do you notice as a pattern& perhaps some gaps developing. How many values did he remove in this round above last round s mean of $112000. How many were above? What do you think will happen to the banker s offer? Any ideas on the bank s offer? How many of you (who haven t already sold back your case would sell it back now?) Are the deals getting more or less  fair ?H  02m.F ? 3380___PPT10.i4@%b20 r(  ^  S Dv     c $c D    h:Hence, do you think the new offer will be higher or lower?;;H  02m.F ? 3380___PPT10.i4@%30  (  ^  S Dv     c $̂ D    ~Ideas on Bank s Offer? Find the percentage of the mean value.H  02m.F ? 3380___PPT10.i4@%40  @(  ^  S Dv     c $+ D    ~Ideas on Bank s Offer? Find the percentage of the mean value.H  02m.F ? 3380___PPT10.i4@%p50 P(  ^  S Dv     c $ D    vbDespite the mean being reduced by more than half, the new offer is 49% of the new mean? What does this say about our conjectures on fairness as the game continues. Who would keep going? Who would take this offer and stop? What has changed now (looks at the board& do you notice patterns such as a gap.) H  02m.F ? 3380___PPT10.i4@%c60 ps(  ^  S Dv     c $ٶ D    iUNow, what do you think the bank will offer will be? Who would stop now? Keep going? H  02m.F ? 3380___PPT10.i4@%c70 s(  ^  S Dv     c $t D    iUNow, what do you think the bank will offer will be? Who would stop now? Keep going? H  02m.F ? 3380___PPT10.i4@%c80  s(   ^   S Dv      c $ D    iUNow, what do you think the bank will offer will be? Who would stop now? Keep going? H   02m.F ? 3380___PPT10.i4@%=90 (M(  (^ ( S Dv    ( c $Lo D    CLast time the offer jumped to $68000, nearly 85% of the mean value. How about now? Any thoughts? Which of you would quit now? Is there anyone left in the audience who is still going? The offer jumped to $90000, nearly 90% of the mean value of the remaining amounts, up from $68000, nearly 85% of the mean value of the remaining amounts before. Even if he eliminates the smallest amount and receives an offer of 100% of the mean, how much will the next offer be? Is the risk worth the reward? 6H ( 02m.F ? 3380___PPT10.i4@%:0 4 (  4^ 4 S Dv    4 c $< D     H 4 02m.F ? 3380___PPT10.i4@%U;0 <e(  <^ < S Dv    < c $ľ D    [GWhat do you think of the offer? What has changed in terms of fairness?H < 02m.F ? 3380___PPT10.i4@%p<0 0D(  D^ D S Dv    D c $Ħ D    vbAsk. Wait. Invite participants to explain how they arrived at their number (even if it is wrong). H D 02m.F ? 3380___PPT10.i4@%J=0 LZ(  L^ L S Dv    L c $( D    P<During a special presentation of  Deal or No Deal, the top prize value was changed from $1 million to $2 million while the other 25 values remained the same H L 02m.F ? 3380___PPT10.i4@%>0 D<T(  T^ T S Dv   6 T c $D9 D    nDuring a special presentation of  Deal or No Deal, the top prize value was changed from $1 million to $2 million while the other 25 values remained the same Changing 1/26 (or 4%) of the values changes the mean by 29%. Why? Which measure of center is more robust (resistant to change due to extreme values?) @8D .H T 02m.F ? 3380___PPT10.i4@%4?0 \D(  \^ \ S Dv    \ c $ D    : H \ 02m.F ? 3380___PPT10.i4@%A0  t(  t^ t S Dv     t c $Dy D    hWhat if the 7 smallest values were doubled, not the 7 largest values? 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Fair or Not Fair?jgershCharlie Burrus31Microsoft PowerPoint@H$@<~B@p9=e G g  g  y--$xx--'@Times New Roman-. 2 +-Deal or No Deal?."System`;-@Times New Roman-. !2 6/Fair or Not Fair?n.-@Times New Roman-. "2 K4Dr. Jason Gershman.-@Times New Roman-. F2 V*Rice University School Mathematics Project.-@Times New Roman-. g2 g@This presentation is property of Jason Gershman and the Rice Uni.-@Times New Roman-. 2 gjversity School ?.-@Times New Roman-. g2 j@Mathematics Project. This project is in no way affiliated with .-@Times New Roman-. 2 jfNBC/Universal, ?.-@Times New Roman-. 2 jzEndemol.-@Times New Roman-. 2 nUSA, o.-@Times New Roman-.  2 n.-@Times New Roman-. 2 n Deal or No Deal.-@Times New Roman-.  2 n4.-@Times New Roman-. N2 n6/or any of its affiliated companies or sponsors..-՜.+,0H    On-screen ShowRice Universityp!! &Times New Roman WingdingsDefault DesignMicrosoft Office Excel ChartMicrosoft Excel Chart#Deal or No Deal? Fair or Not Fair?Deal or No DealThe Game BoardThe Game Board AgainThe Game BeginsGame Board at 1st DealGame Board at 1st DealGame Board at 2nd DealGame Board at 2nd DealGame Board at 3rd DealGame Board at 3rd DealGame Board at 3rd DealFairness Offer : Mean RatioGame Board at 4th DealGame Board at 4th DealGame Board at 4th DealGame Board at 5th Deal ObservationsGame Board at 5th DealGame Board at 6th DealGame Board at 6th DealGame Board at 7th DealGame Board at 7th DealSome ThoughtsGame Board at 8th DealGame Board at 8th Deal Fairness ConclusionsNew Game BoardNew Game BoardPremier Game BoardPremier Game BoardAlternative Game Board  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles!&_ůCharlie BurrusCharlie Burrus  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)Pictures0Current UserSummaryInformation(PowerPoint Document( DocumentSummaryInformation8