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

1. 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

2. 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.

3. Okay, so what this is actually saying then is that it cannot include an intersection of the two whatsoever?

4. 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.

5. 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
$\begin{gathered}
P\left( {A \cap \overline B } \right) + P\left( {\overline A \cap B} \right) \hfill \\ P\left( A \right) - P\left( {A \cap B} \right) + P\left( B \right) - P\left( {A \cap B} \right) \hfill \\ P\left( A \right) + P\left( B \right) - 2P\left( {A \cap B} \right) \hfill \\ \end{gathered}$

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exactly one of the two events occur

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