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Math Help - Proof union disjoint events

  1. #1
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    Proof union disjoint events

    Proofs scare the living daylights out of me! I could really use some help with this:

    For some experiment, suppose B and C are events. The event, D, that "exactly one of the events B and C occur" is of interest.
    (a) Show that D can be written as a union of disjoint events.
    (b) Show that P(D) = P(B) + P(C) -2P(B intersect C).

    Thanks!
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  2. #2
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    Gdansk, Poland
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    (a)

    D = (B \setminus C) \cup (C \setminus B)

    (b)

    We can use the above decomposition to do that, however I think it is easier to do this other way.
    We will use the formula, valid for any events A,B:
    P(A \cup B) = P(A) + P(B) - P(A \cap B)

    Hence

    P(D) = P((B \cup C) \setminus (B \cap C)) = P(B \cup C) - P(B \cap C) = P(B) + P(C) - 2P(B \cap C)
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