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Thread: Venn Diagram of A,B and C

  1. #1
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    Venn Diagram of A,B and C

    Can anyone tell me if this is the right Venn Diagram for denoting C the event that exactly one of A and B occurs?

    Its the one on the right at this website:
    Venn Diagram -- from Wolfram MathWorld

    Also how would you write an expression for C in terms of unions, intersections and complement of A and B.

    And then use that to find an expression of P(C) in terms of P(A),P(B) and P(A n B)
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  2. #2
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    C is the event that exactly one of A and B occurs. You only need circles for A and B on the Venn Diagram

    So it means that either:

    A occurs (but not B)
    or
    B occurs (but not A)

    So P(C) = P($\displaystyle A$ n $\displaystyle B^c$) U P($\displaystyle B$ n $\displaystyle A^c$)
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  3. #3
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    Quote Originally Posted by Janu42 View Post
    C is the event that exactly one of A and B occurs. You only need circles for A and B on the Venn Diagram

    So it means that either:

    A occurs (but not B)
    or
    B occurs (but not A)

    So P(C) = P($\displaystyle A$ n $\displaystyle B^c$) U P($\displaystyle B$ n $\displaystyle A^c$)
    So P(C) = P($\displaystyle A$ n $\displaystyle B^c$) U P($\displaystyle B$ n $\displaystyle A^c$)[/QUOTE] would be the expression for C in terms of unions, intersections and complements of A and B.

    How would I use this to derive an expression for P(C) in terms of P(A), P(B) and P(AnB).

    Would it be like this?

    P(C) = ( P(A) - P(A n B) ) u ( P(B) - P(A n B))


    This is the question:

    (a) draw a Venn diagram
    (b) Write down an expression for C in terms of union, intersections and complement of A and B.
    (c) Using result (b) derive an expression for P(C) in terms of P(A), P(B) and P(A nB). Give a detailed proof of the result.


    I dont no what the difference for the expression of C and the expression of P(C) is?
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  4. #4
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    Quote Originally Posted by mathsandphysics View Post
    So P(C) = P($\displaystyle A$ n $\displaystyle B^c$) U P($\displaystyle B$ n $\displaystyle A^c$) would be the expression for C in terms of unions, intersections and complements of A and B.
    How would I use this to derive an expression for P(C) in terms of P(A), P(B) and P(AnB).

    Would it be like this?

    P(C) = ( P(A) - P(A n B) ) u ( P(B) - P(A n B))


    This is the question:
    (c) Using result (b) derive an expression for P(C) in terms of P(A), P(B) and P(A nB). Give a detailed proof of the result.
    $\displaystyle A~\&~B$ are events. So is $\displaystyle C=(A\cap B^c)\cup(B\cap A^c)= A\Delta B$ the symmetric difference of $\displaystyle A~\&~B$.
    There the probability of $\displaystyle C$ depends of the probabilities of $\displaystyle A~\&~B$.

    Thus $\displaystyle P(C)=P(A)+P(B)-2(A\cap B)$
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