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Math Help - probability proof

  1. #1
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    probability proof

    Hi, can anyone please help me with this question? Thanks a lot.

    Let A, B, and C be any three events. Show that:
    (1) P(A)=P(B) if and only if P(A \cap B^c)=P(A^c \cap B)
    (2) Given P(A)=0.5 and P(A \cup (B^c \cap C^c)^c)=0.8, determine P(A^c \cap (B \cup C))

    For first part, I don't know how to start at all. Plus, with "if and only if", does that mean I need to prove from two directions?
    For second part, I did draw Venn Diagram, but still don't feel that is very helpful, except I did find P(A \cup B \cup C)=0.8, am I right? What should I do?
    Thanks a lot.
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  2. #2
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    Quote Originally Posted by tsang View Post
    Let A, B, and C be any three events. Show that:
    (1) P(A)=P(B) if and only if P(A \cap B^c)=P(A^c \cap B)
    For any two events this is true:
    P(A)=P(A\cap B)+(A\cap B^c) and
    P(B)=P(B\cap A)+(B\cap A^c).

    Use that to prove:
    1) If P(A)=P(B) then P(A \cap B^c)=P(A^c \cap B).
    and
    2) If P(A \cap B^c)=P(A^c \cap B) then P(A)=P(B).
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  3. #3
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    Quote Originally Posted by tsang View Post
    (2) Given P(A)=0.5 and P(A \cup (B^c \cap C^c)^c)=0.8, determine P(A^c \cap (B \cup C))
    For second part, I did draw Venn Diagram, but still don't feel that is very helpful, except I did find P(A \cup B \cup C)=0.8, am I right?.
    That is correct. Now find the area shaded in yellow.
    Attached Thumbnails Attached Thumbnails probability proof-untitled.gif  
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  4. #4
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    Thanks a lot Plato. I need to keep trying on part 1, it is still a bit hard for me. If I still can't get it, I may ask you for help again. Thank you.
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  5. #5
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    Quote Originally Posted by tsang View Post
    If I still can't get it, I may ask you for help again. Thank you.
    You cannot apply  P(B\cap A)+(B\cap A^c)=P(A\cap B)+(A\cap B^c) and subtract?
    What gives here?
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  6. #6
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    Quote Originally Posted by Plato View Post
    You cannot apply  P(B\cap A)+(B\cap A^c)=P(A\cap B)+(A\cap B^c) and subtract?
    What gives here?
    Thank you very much, Plato, sorry I was just thinking too much, because I did too much Real Analysis last year, so when I saw that "if and only if" statement, I got scared and thought I need to write heaps of detailed proof. So yes, you are absolutely right, it was just me thinking too much.

    May I ask about part 2 again, I calculated the result is just 0.3, in fact can just use 0.8-0.5=0.3, but the only thing is could you please give me advice on how to write about it? Because I feel only use Venn diagram to show is not quite mathematical, could you please tell me if there is anyway to write about it? I did get A U B U C=0.8, but that is really because I worked out from the diagram. Do you think I can do it algebracally? Thanks again.
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