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Find expression for event exactly one of A,B,C occurs

  1. Aug 8, 2012 #1
    Find expression for event "exactly one of A,B,C occurs"

    1. The problem statement, all variables and given/known data

    Let [itex]A,B,C \in F[/itex] be three arbitrary events. F is a σ-algebra. Find an expression for the event "exactly one of A,B,C occurs".

    3. The attempt at a solution

    Define [itex]i,j,k[/itex] s.t. [itex]i,j,k \in \{1,2,3\} \wedge i \not= j \not=k[/itex]. Also, define [itex]M_1 := A, M_2 := B, M_3 := C[/itex].

    Then the event in question is [itex]M_i \backslash (M_j \cup M_k) \forall i[/itex].

    -------------

    Is this correct? I've never seen this "notation" used in my lecture slides so I'm not sure (I'm trying to get out of writing a silly amount of unions and then simplifying).
     
    Last edited: Aug 8, 2012
  2. jcsd
  3. Aug 8, 2012 #2

    Ray Vickson

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    Re: Find expression for event "exactly one of A,B,C occurs"

    Yes, [itex] A\text{ only} = A \cap (B \cup C)^c,[/itex] etc, where [itex] D^c[/itex] denotes the complement of a set D. As to the probability that exactly one occurs, see Feller, "Introduction to Probability Theory and its Applications", Vol I, (Wiley), which gives expressions for P{exactly k events occur} for k = 0, 1, 2, ... among n events. You can also find similar developments in the various Probability books by Sheldon Ross.

    RGV
     
  4. Aug 8, 2012 #3
    Re: Find expression for event "exactly one of A,B,C occurs"

    Thanks, I'll borrow that book from my uni's library.

    So what I've done is correct?
     
  5. Aug 8, 2012 #4
    Re: Find expression for event "exactly one of A,B,C occurs"

    Actually I believe that I've made a mistake in my OP.

    The correct notation for the event would be [itex]\bigcup_{i=1}^3 M_i \backslash (M_j \cup M_k)[/itex] for [itex]i \not= j \not= k[/itex] and [itex]j,k \in \{1,2,3\}[/itex].

    Still would like confirmation that this is correct :).
     
    Last edited: Aug 8, 2012
  6. Aug 8, 2012 #5

    Ray Vickson

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    Re: Find expression for event "exactly one of A,B,C occurs"

    It is obviously true.

    RGV
     
  7. Aug 8, 2012 #6
    Re: Find expression for event "exactly one of A,B,C occurs"

    Thanks for your help. Just wasn't sure of the notation :)
     
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