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Math Help - probability--proving that exactly one of two events will happen

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    probability--proving that exactly one of two events will happen

    evening

    how do I show that the probability that exactly one of the events A or B occurs equals P(A) + P(B) - 2*P(AB)?

    i know that i'm supposed to use the axioms of probability, but i don't know where to start
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    Quote Originally Posted by pittsburghsteelers View Post
    evening

    how do I show that the probability that exactly one of the events A or B occurs equals P(A) + P(B) - 2*P(AB)?

    i know that i'm supposed to use the axioms of probability, but i don't know where to start
    Start by drawing a Venn diagram.
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    Okay, so what this is actually saying then is that it cannot include an intersection of the two whatsoever?
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    Quote Originally Posted by pittsburghsteelers View Post
    Okay, so what this is actually saying then is that it cannot include an intersection of the two whatsoever?
    Yes. Because the intersection means both events occur.
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    Quote Originally Posted by pittsburghsteelers View Post
    how do I show that the probability that exactly one of the events A or B occurs equals P(A) + P(B) - 2*P(AB)?
    For any events X&Y it is true that
    P\left( X \right) = P\left( {X \cap Y} \right) + P\left( {X \cap \overline Y } \right)\, \Rightarrow \,P\left( {X \cap \overline Y } \right) = P\left( X \right) - P\left( {X \cap  Y } \right)

    Using that we get
    \begin{gathered}<br />
  P\left( {A \cap \overline B } \right) + P\left( {\overline A  \cap B} \right) \hfill \\  P\left( A \right) - P\left( {A \cap B} \right) + P\left( B \right) - P\left( {A \cap B} \right) \hfill \\  P\left( A \right) + P\left( B \right) - 2P\left( {A \cap B} \right) \hfill \\ \end{gathered}
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