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Math Help - events

  1. #1
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    events

    events A and B are such that
    p(A) = 1/2
    p(A'|b) = 1/3
    p(A \cup B) = 3/5

    determine P(B|A').
    totally have no idea..
    how can i define p(A \cupB) = 3/5 btw?
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  2. #2
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    Quote Originally Posted by z1llch View Post
    events A and B are such that
    p(A) = 1/2
    p(A'|b) = 1/3
    p(A \cup B) = 3/5

    determine P(B|A').
    totally have no idea..
    how can i define p(A \cupB) = 3/5 btw?
    General probability rules for your question...

    P(A') = 1 - P(A)

    P(A \cup B) = P(A)+P(B)-P(A\cap B)

    P(A|B) = \frac{P(A\cap B)}{P(B)}
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  3. #3
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    Quote Originally Posted by z1llch View Post
    events A and B are such that
    p(A) = 1/2
    p(A'|b) = 1/3
    p(A \cup B) = 3/5

    determine P(B|A').
    totally have no idea..
    how can i define p(A \cupB) = 3/5 btw?
    I'd suggest you draw a Karnaugh table. From the given data:


    \begin{tabular}{l | c | c | c} & A & A$\, '$ & \\ \hline B & a & b & \\ \hline B$\, '$ & c & (1/2) - b & \\ \hline & 1/2 & 1/2 & 1 \\ \end{tabular}<br />

    where:

    a + b + c = \frac{3}{5} .... (1)

    \frac{b}{a+b} = \frac{1}{3} \Rightarrow a = 2b .... (2)

    Substitute (2) into (1): c = \frac{3}{5} - 3b.

    Update the Karnaugh table:


    \begin{tabular}{l | c | c | c} & A & A$\, '$ & \\ \hline B & 2b & b & 3b \\ \hline B$\, '$ & (3/5) - 3b & (1/2) - b & (11/10) - 4b \\ \hline & 1/2 & 1/2 & 1 \\ \end{tabular}<br />


    Therefore 3b + \frac{11}{10} - 4b = 1 \Rightarrow b = \frac{1}{10}.

    Update the Karnaugh table:


    \begin{tabular}{l | c | c | c} & A & A$\, '$ & \\ \hline B & 2/10 & 1/10 & 3/10 \\ \hline B$\, '$ & 3/10 & 4/10 & 7/10 \\ \hline & 1/2 & 1/2 & 1 \\ \end{tabular}<br />


    Now it's easy to get \Pr(B | A') = \frac{\Pr(B \cap A')}{\Pr(A')} = \frac{1/10}{1/2} = \frac{1}{5}.
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  4. #4
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    that karnaugh table was great! i don't think it's in my syllabus though learnt a new thing, thanks a lot
    the question suggested venn diagram, but i sucked in that. can we 'possibly' just do it with general rules lol?
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