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45-733 PROBABILITY AND STATISTICS I


Answer to Problem 2


Let A1 = Urn I, A2 = Urn II,
  1. B = Yellow Ball is Drawn, Bc = 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)]/ [åj=1,2 P(Aj)P(B|Aj)] = 
    
                         1 10
                         -*--
                         2 30       5
         P(A2 | B) = ----------- = --
                     1 16   1 10   13
                     -*-- + -*--
                     2 30   2 30


  2. B = Red Ball is Drawn, Bc = 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)]/ [åj=1,2 P(Aj)P(B|Aj)] = 
    
                         1  2
                         -*--
                         2 30      1
         P(A1 | B) = ----------- = -
                     1  2   1  8   5
                     -*-- + -*--
                     2 30   2 30