45-733 PROBABILITY AND STATISTICS I

Answer to Problem 2

1. B = Yellow Ball is Drawn, B^c = Green or Red Ball is Drawn
   
   A Diagram of this is:
   
   P(A₁) = P(A₂) = 1/2
   P(B | A₁) = 16/30
   P(B | A₂) = 10/30
   Hence:
   
   P(A₂|B) = [P(A₂)P(B|A₂)]/ [å_j=1,2 P(A_j)P(B|A_j)] =

   
                            1 10
                            -*--
                            2 30       5
            P(A2 | B) = ----------- = --
                        1 16   1 10   13
                        -*-- + -*--
                        2 30   2 30
   

   
   
   
2. B = Red Ball is Drawn, B^c = Yellow or Green Ball is Drawn
   
   A Diagram of this is:
   
   P(A₁) = P(A₂) = 1/2
   P(B | A₁) = 2/30
   P(B | A₂) = 8/30
   Hence:
   
   P(A₁|B) = [P(A₁)P(B|A₁)]/ [å_j=1,2 P(A_j)P(B|A_j)] =

   
                            1  2
                            -*--
                            2 30      1
            P(A1 | B) = ----------- = -
                        1  2   1  8   5
                        -*-- + -*--
                        2 30   2 30