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[Advanced] Probability of Union[n-elements]

  1. Oct 6, 2011 #1
    Hello,

    We are all familiar with the formula that relates union of 2 mutually NOT exclusive events formula:

    [itex] P(A\cup B)=P(A)+P(B)-P(A\cap B) [/itex]

    For 3 sets its easily derived using this formula.

    But I wanted to take this step further. I wanted to find a general formula, that represents union of n elements.

    I don't know how to write that In LaTex.

    If anybody knows the answer, please don't tell me. Tell me some guidelines to solution. I have tried, but I get stuck with recursive sums, and I can't get out of them.
     
  2. jcsd
  3. Oct 6, 2011 #2
    Think of expressing a union of n elements as a union of a smaller number of elements,and then use the answer you already know, by using parentheses. I am trying to not be neither too obscure nor tell you the answer.
     
  4. Oct 6, 2011 #3
    You mean like, doing for 3 4 5 and maybe 6 sets this union, then get my answer from that?

    Is that mathematically bulletproof?
     
  5. Oct 6, 2011 #4

    micromass

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    It's certainly not bulletproof, but it's a nice start. Start by finding it for 3,4,5 and 6 and see if you can generalize it. Once you've found a candidate for a general solution, then you can apply induction to prove it.
     
  6. Oct 6, 2011 #5
    Fun! On it
     
  7. Oct 6, 2011 #6
    Here is what I got so far:

    [itex]P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B\cap C)+P(A\cap B\cap C)[/itex]

    Assuming that:

    [itex] P((A\cap B\cap C)\cup D)=P((A\cap D)\cup (B\cap D)\cup (C\cap D) [/itex]

    then:

    [itex]P(A\cup B\cup C\cup D)=P(A)+P(B)+P(C)+P(D)-P(A\cap B)-P(A\cap C)-P(A\cap D)-P(B\cap C)-P(B\cap D)-P(C\cap D)
    +P(A\cap B\cap C)+P(A\cap C\cap D)+P(B\cap C\cap D)-P(A\cap B\cap C\cap D)[/itex]


    Ok, I see a sum here [itex]\sum_{i=1}^{n} P(A_{i})[/itex]

    I also see that each set is intersected with every other set. I don't know exactly how to write that.

    At least not in LaTex. I am thinking:

    [itex] \bigcap_{i,j=1}^{n} A_{i},A_{j} i\neq j[/itex] [idea]

    I get stuck at those 3 intersections and 4.
     
  8. Oct 6, 2011 #7
    Now, try to see if you can detect a pattern and try induction. If you want, I can give you the (an) answer with a spoiler warning, for when you're done' let me know.
     
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