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AA BB CC DD EE FF GG HH II JJ KK LL MM NN OOPrepared  by: Mike  PangburnDate:   Feb.  6,  2013Question  1

One  play  of  game12  players'  choices69396748686719078949288

#  unique  values? 11Was  there  match? Yes

Simulate  1000  plays  (trials)  using  a  Data  TableMatch?

Yes1 Yes2 No3 No4 No5 No6 No7 No8 No9 No

995 No996 No997 No998 No999 No1000 Yes

%  of  games  with  match? 48%

Question  2

One  play  of  game

Player  A's  visible  die 6Player  A's  hidden  die 4

Player  B  decides  how  many  dice  to  roll:  1,  2,  or  3.    Let's  call  these  Options  a,  b,  c.Player  B's  subsequent  dice  (not  all  will  apply-­‐-­‐unless  she  chose  Option  3)Die  1 4Die  2 2Die  3 5

Player  A's  sum 10 (Note:  this  is  unknown  to  Player  B  until  after  she  chooses  between  Options  a,b,c)Relevant  player  B  sums  with  Option…Option  a Option  b Option  c

4 6 11

Would  player  B  win? No No No

Simulate  many  plays  of  gameScenario:  A's  visible  die  is  a  1 Scenario:  A's  visible  die  is  a  2 Scenario:  A's  visible  die  is  a  3

1000  trials  below No No No 1000  trials  below No No No 1000  trials  below No No No1 No No No 2 No No No 3 No Yes No1 No No No 2 No No No 3 No No No1 No No No 2 No No No 3 No No No1 No No No 2 Yes No No 3 No No No1 No No No 2 Yes No No 3 No No No1 No No No 2 Yes No No 3 No No No1 No No No 2 Yes No No 3 No Yes No1 No No No 2 No No No 3 No Yes No1 No No No 2 No No No 3 No No Yes1 No Yes No 2 No No No 3 Yes No No1 Yes No No 2 No No No 3 No No No1 No No No 2 No No No 3 No No No1 No No No 2 No No No 3 No No No1 Yes No No 2 No No No 3 No No No

%  games  player  B  won? 28% 10% 3% %  games  player  B  won? 17% 12% 6% %  games  player  B  won? 8% 13% 10%

Scenario:  A's  visible  die  is  a  4 Scenario:  A's  visible  die  is  a  5 Scenario:  A's  visible  die  is  a  61000  trials  below No No No 1000  trials  below No No No 1000  trials  below No No No

4 No No No 5 No Yes No 6 No No No4 No No No 5 No No No 6 No No No4 No No No 5 No No Yes 6 No Yes No4 No No No 5 No No No 6 No No No4 No No Yes 5 No Yes No 6 No No No4 No No No 5 No No Yes 6 No Yes No4 No No No 5 No No Yes 6 No No Yes4 No No No 5 No No Yes 6 No No No4 No No No 5 No No No 6 No No Yes4 No No No 5 No No No 6 No No No4 No No Yes 5 No No No 6 No No No4 No No No 5 No No No 6 No No Yes

%  games  player  B  won? 3% 13% 19% %  games  player  B  won? 0% 13% 25% %  games  player  B  won? 0% 10% 34%

Therefore,  based  on  the  results  you  find  above,  what  will  your  strategy  be?  I  will  choose  to  roll  2  dice  whenever  I  see  my  opponent's  first  roll  is:I  will  choose  to  roll  1  die  whenever  I  see  my  opponent's  first  roll  is:I  will  choose  to  roll  3  dice  whenever  I  see  my  opponent's  first  roll  is:

(Remember  here  that  winning  with  2  dice  gets  you  2  points,  but  winning  with  1  or  3  dice  gets  you  only  1  point.)

Your  opponent's  2nd  random  die's  value  is  here,  between  1  and  6,  but  you  don't  get  to  see  it  uncl  you  decide  how  many  dice  you  will  roll  (to  emphasize  that,  I  colored  the  cell  black).    For  this  parccular  trial,  we  can  see  from  cell  B1039  (=B1030+B1031)  that  this  value  must  be  4