20-CS-110-001 Introduction to Computer Science Fall 2010

Solution to Dashing Courtier Puzzle

   
Pr(courtier gets married) =

  Pr(courtier gets married and he picks the doors with 2 tigers  OR
     courtier gets married and he picks the doors with 1 tiger  OR
     courtier gets married and he picks the doors with 0 tigers) =

  Pr(courtier gets married and he picks the doors with 2 tigers) +
     courtier gets married and he picks the doors with 1 tiger) +
     courtier gets married and he picks the doors with 0 tigers)

  Because the events separated by "OR" are mutually exclusive
	 
  Pr(courtier gets married and he picks the doors with 2 tigers) = 0
  
  Pr(courtier gets married and he picks the doors with 0 tigers) =
    Pr(courtier gets married given he picks the doors with 0 tigers)*
    Pr(courtier picks the doors with 0 tigers) = 1*(1/3) = 1/3

  Pr(courtier gets married and he picks the doors with 1 tiger) =
    Pr(courtier gets married given he picks the doors with 1 tiger)*
    Pr(courtier picks the doors with 1 tiger) = (1/8)*(1/3) = 1/24
	 
  since
	 
    Pr(courtier picks the doors with 1 tiger) = 1/3
    Pr(courtier gets married given he picks the doors with 1 tiger) = 
       Pr(courtier picks the door with the lady 3 times) = (1/2)^3 = 1/8
		 
  Hence
  Pr(courtier gets married) = 0 + 1/3 + 1/24 = 9/24 = 3/8

  ++  ++  ++  ++  ++  ++  ++  ++  ++  ++  ++  ++  ++  ++  ++  ++  ++  ++

  Pr(courtier gets married given first two choices are ladies) = 
    Pr(courtier gets married AND first two choices are ladies)/
    Pr(first two choices are ladies)

    Pr(first two choices are ladies) =
      Pr(first two choices are ladies and chose 0 tigers) +
      Pr(first two choices are ladies and chose 1 tiger) = 
      (1/3) + (1/3)*(1/4) = 5/12

    Pr(courtier gets married AND first two choices are ladies) =
    Pr(courtier gets married) = 3/8 (from above)

  Hence
    Pr(courtier gets married given first two choices are ladies) = 
      (3/8)/(5/12) = 36/40 = 9/10