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Probability Question

  1. Feb 5, 2013 #1
    1. The problem statement, all variables and given/known data

    Suppose
    P(A|B) = P(A|B^c)
    (P(B) > 0 and P(B^c) > 0 both understood). Show that A and B are independent.

    2. Relevant equations



    3. The attempt at a solution

    I don't know where to go from here. Thanks for any help

    [itex]\frac{P(A \bigcap B)}{P(B)} = \frac{P(A \bigcap B^{c})}{P(B^{c})}[/itex]
     
    Last edited: Feb 5, 2013
  2. jcsd
  3. Feb 5, 2013 #2

    HallsofIvy

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    Please restate (in fact, go back are reread) the problem. As you have it now it says "Suppose A and B are independent. Show that A and B are independent"!
     
  4. Feb 5, 2013 #3
    Ok sorry about that I fixed that. Thanks for helping me.
     
  5. Feb 5, 2013 #4

    HallsofIvy

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    Okay, good. Now what is your definition of "independent events"?
     
  6. Feb 5, 2013 #5
    P(A intersection B) = P(A)P(B)

    not sure what to do from here because it's conditional probabilities.
     
  7. Feb 5, 2013 #6

    HallsofIvy

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    Good! So you are given that [itex]P(A|B)= P(A|B^c)[/itex] and want to prove that [itex]P(A\cap B)= P(A)P(B)[/itex].

    You should know, then, that [itex]P(A)= P(A|B)P(B)+ P(A|B^c)B^c[/itex], for any A and B. Further, [itex]P(B^c)= 1- P(B)[/itex].

    (The reason I asked about the definition was that some texts use "P(A)= P(A|B)" as the definition of "A and B are independent. Of course it is easy to show that is equivalent to your "[itex]P(A\cap
    B)= P(A)P(B)[/itex]".)
     
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