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

- Jan 19th 2009, 04:27 PMpittsburghsteelersprobability--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 - Jan 19th 2009, 04:32 PMmr fantastic
- Jan 19th 2009, 05:39 PMpittsburghsteelers
Okay, so what this is actually saying then is that it cannot include an intersection of the two whatsoever?

- Jan 19th 2009, 07:30 PMmr fantastic
- Jan 20th 2009, 03:34 AMPlato
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} $