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

• Jan 19th 2009, 04:27 PM
pittsburghsteelers
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
• Jan 19th 2009, 04:32 PM
mr fantastic
Quote:

Originally Posted by pittsburghsteelers
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.
• Jan 19th 2009, 05:39 PM
pittsburghsteelers
Okay, so what this is actually saying then is that it cannot include an intersection of the two whatsoever?
• Jan 19th 2009, 07:30 PM
mr fantastic
Quote:

Originally Posted by pittsburghsteelers
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.
• Jan 20th 2009, 03:34 AM
Plato
Quote:

Originally Posted by pittsburghsteelers
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
$\begin{gathered}
P\left( {A \cap \overline B } \right) + P\left( {\overline A \cap B} \right) \hfill \\ P\left( A \right) - P\left( {A \cap B} \right) + P\left( B \right) - P\left( {A \cap B} \right) \hfill \\ P\left( A \right) + P\left( B \right) - 2P\left( {A \cap B} \right) \hfill \\ \end{gathered}$