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Independent events in probabilities

  1. Mar 15, 2010 #1
    1. The problem statement, all variables and given/known data
    Let S be the sample space for rolling a single die. Let A={1,2,3,4}, B={2,3,4}, and C={3,4,5}. Which of the pairs (A,B), (A,C), and (B,C) is independent?


    2. Relevant equations
    P(A|B)=P(A)
    P(A|B)=P(A&B)/P(B)
    P(A&B)=P(A)*P(B)

    3. The attempt at a solution
    P(A)=2/3 P(B)=1/2 P(C)=1/2
    P(A&B)=P(A)*P(B)=(2/3)(1/2)=1/3 P(A&B)/P(B)=(1/3)/(1/2)=1/6
    P(A&C)=1/3 P(A&C)/P(C)=(1/3)/(1/2)=1/6
    P(B&C)=1/4 P(B&C)/P(C)=(1/4)/(1/2)=1/8

    So based on my calculations, there is none of the pairs which match the independence rule. But the book says that (A,C) is independent.
     
  2. jcsd
  3. Mar 15, 2010 #2

    vela

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    The first and third equations only hold when A and B are independent. The second equation holds generally.
    You can't ignore the outcomes included in each set. For example, if events A and B are both true, that means that you rolled a 2, 3, or 4, so P(A&B)=1/2.
     
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