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Math Help - Help me find P(A) and P(B)

  1. #1
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    Help me find P(A) and P(B)

    Hey all!

    GIVEN:
    P(A|B) = 0.8
    P(B|A) = 0.5
    P(AuB) = 0.9

    The question:
    Find P(A) and P(B).

    My attempt:
    I've done many questions like this, however they have some substantial data in them and it's easy to get the general feel on how to solve them. This question however I cannot solve at all.

    My biggest problem is that I do not understand what to do with P(AuB) if I dont know weather P(A) and P(B) are independent events; P(AuB) = P(A)*P(B). How do I solve this without that assumption? I have no laws about it in my book and the three I do have just gets me in a circle and back to nothing. Please help!
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Help me find P(A) and P(B)

    Use the definition of a conditional probability.
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  3. #3
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    Re: Help me find P(A) and P(B)

    P(A|B) = P(AnB)/P(B) ? Then I dont have P(AnB).. and P(AnB) = P(A) + P(B) - P(AuB). Trying many ways I still have three unknowns and two equations. Is there something i've missed?
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  4. #4
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    Re: Help me find P(A) and P(B)

    Quote Originally Posted by Hanga View Post
    P(A|B) = P(AnB)/P(B) ? Then I dont have P(AnB).. and P(AnB) = P(A) + P(B) - P(AuB). Trying many ways I still have three unknowns and two equations. Is there something i've missed?
    I agree that something is missing here.
    Are you sure that you were asked to find P(A)~\&~P(B)~?
    Could you have been asked to find bounds on those values?

    Why not double check the wording?
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  5. #5
    MHF Contributor Siron's Avatar
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    Re: Help me find P(A) and P(B)

    modified, sorry I made a mistake in it, my apologies.
    I agree with Plato.
    Last edited by Siron; August 22nd 2011 at 12:03 PM.
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  6. #6
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    Re: Help me find P(A) and P(B)

    The exact wordings are:

    For two events A and B we have P(A|B) = 0.8, P(B|A) = 0.5, P(AuB)=0.9
    Decide P(A) and P(B).

    I'm still trying to figure this out but no dice
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  7. #7
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    Re: Help me find P(A) and P(B)

    Quote Originally Posted by Hanga View Post
    The exact wordings are:

    For two events A and B we have P(A|B) = 0.8, P(B|A) = 0.5, P(AuB)=0.9
    Decide P(A) and P(B).

    I'm still trying to figure this out but no dice
    Dice are irrelevant here.

    Pr(A) = 0.8 Pr(B) .... (1)

    Pr(B) = 0.5 Pr(A) .... (2)

    Solve equations (1) and (2) simultaneously for Pr(A) and Pr(B) etc.
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  8. #8
    MHF Contributor Siron's Avatar
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    Re: Help me find P(A) and P(B)

    My attempt (sorry for my previous post):
    P(A|B)=\frac{P(A\cap B)}{P(B)}
    P(B|A)=\frac{P(B\cap A)}{P(A)}
    And P(A|B)\neq P(B|A) whereas P(A\cap B)=P(B\cap A) and therefore:
    P(A|B)\cdot P(B)=P(B|A)\cdot P(A)

    If we enter the given information we get:
    0,8P(B)=0,5P(A) \Leftrightarrow P(A)=\frac{8}{5}P(B) \Leftrightarrow P(B)=\frac{5}{8}P(A)

    We know also this formula:
    P(A\cap B)=P(A)+P(B)-P(A\cup B)
    If we enter the given information we get:
    P(A|B)\cdot P(B)= P(A)+P(B) - P(A\cup B)
     0,8\cdot \frac{5}{8}P(A)=P(A)+\frac{5}{8}P(A)-0,9
    P(A)=\frac{4}{5}
    and so P(B)=\frac{1}{2}
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  9. #9
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    Re: Help me find P(A) and P(B)

    Quote Originally Posted by mr fantastic View Post
    Dice are irrelevant here.

    Pr(A) = 0.8 Pr(B) .... (1)

    Pr(B) = 0.5 Pr(A) .... (2)

    Solve equations (1) and (2) simultaneously for Pr(A) and Pr(B) etc.
    Errr...hemm..

    Pr(AnB)=Pr(A|B)Pr(B)=Pr(B|A)Pr(A)

    which means you have enough information to determine the ratio of Pr(A) and Pr(B). So Pr(B)=(0.5/0.8)Pr(A)

    0.5=Pr(AuB)=Pr(A)+Pr(B)-Pr(AnB)=Pr(A)(1+5/8)-Pr(B|A)Pr(A)

    etc
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  10. #10
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    Re: Help me find P(A) and P(B)

    I just could not clear my brain enough to see through this problem.
    But now I think it is real interesting.

    Given P(A|B),~P(B|A)~\&~P(A\cup B) then

     \begin{align*}P(A)&=\frac{P(A|B)\cdot P(A\cup B)}{P(A|B)+P(B|A)-P(A|B)\cdot P(B|A)}\\ P(B) &=\frac{P(B|A)\cdot P(A\cup B)}{P(A|B)+P(B|A)-P(A|B)\cdot P(B|A)} \end{align*}
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  11. #11
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    Re: Help me find P(A) and P(B)

    Thank you very much guys! I'm going to study the method today and make sure I understand it perfectly, I belive the key was P(A|B)P(B)=P(B|A)P(A) and some smart moves on your part!
    Have a nice day!
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  12. #12
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    Re: Help me find P(A) and P(B)

    Quote Originally Posted by CaptainBlack View Post
    Errr...hemm..

    Pr(AnB)=Pr(A|B)Pr(B)=Pr(B|A)Pr(A)

    which means you have enough information to determine the ratio of Pr(A) and Pr(B). So Pr(B)=(0.5/0.8)Pr(A)

    0.5=Pr(AuB)=Pr(A)+Pr(B)-Pr(AnB)=Pr(A)(1+5/8)-Pr(B|A)Pr(A)

    etc
    Yes, I'll have to learn to post only when I'm awake.
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  13. #13
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    Re: Help me find P(A) and P(B)

    i too got the same answer....
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