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
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
For any events X&Y it is true that
$\displaystyle P\left( X \right) = P\left( {X \cap Y} \right) + P\left( {X \cap \overline Y } \right)\, \Rightarrow \,P\left( {X \cap \overline Y } \right) = P\left( X \right) - P\left( {X \cap Y } \right)$
Using that we get
$\displaystyle \begin{gathered}
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} $