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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  ?l