Probability Question

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Homework Statement



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.

Homework Equations





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]
 
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Answers and Replies

  • #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"!
 
  • #3
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Ok sorry about that I fixed that. Thanks for helping me.
 
  • #4
HallsofIvy
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Okay, good. Now what is your definition of "independent events"?
 
  • #5
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P(A intersection B) = P(A)P(B)

not sure what to do from here because it's conditional probabilities.
 
  • #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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