Show that if events A and B are independent, then events A' and B are independent.

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- Jan 11th 2009, 01:03 PMYanshow that events are independent
Show that if events A and B are independent, then events A' and B are independent.

- Jan 11th 2009, 04:56 PMLast_Singularity
If events $\displaystyle A,B$ are independent, then we know that

$\displaystyle P(A \cap B) = P(A) P(B)$ by definition. That's our key.

For the proof, we will use the following lemma:

$\displaystyle A^C \cap B = B \setminus (A \cap B)$

Proof of the lemma:

$\displaystyle A^C \cap B = (\Omega \setminus A) \cap B = (\Omega \cap B) \setminus (A \cap B) = B \setminus (A \cap B)$

So now we know that lemma is true, we can use it to conclude that:

$\displaystyle P(A^C \cap B) = P(B) - P(A \cap B)$

Looking at the right hand side, we have:

$\displaystyle =P(B) - P(A \cap B)$

$\displaystyle =P(B) - P(A)P(B)$ (using our key assumption)

$\displaystyle =(1-P(A))P(B)$

$\displaystyle =P(A^C)P(B)$

So we conclude that

$\displaystyle P(A^C \cap B) = P(A^C)P(B)$ as desired.

$\displaystyle QED$ - Jan 11th 2009, 05:31 PMPlato
$\displaystyle \begin{array}{*{20}c}

{P\left( {A^c \cap B} \right)} & = & {P\left( B \right) + P\left( {A \cap B} \right)} \\

{} & = & {P\left( B \right) + P\left( A \right)P\left( B \right)} \\

{} & = & {P(B)\left( {1 - P(A)} \right)} \\

{} & = & {P(B)P\left( {A^c } \right)} \\

\end{array} $ - Jan 11th 2009, 05:35 PMmr fantastic