Thread: show that events are independent

1. show that events are independent

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

2. Originally Posted by Yan
Show that if events A and B are independent, then events A' and B are independent.
If events $A,B$ are independent, then we know that
$P(A \cap B) = P(A) P(B)$ by definition. That's our key.

For the proof, we will use the following lemma:
$A^C \cap B = B \setminus (A \cap B)$

Proof of the lemma:
$A^C \cap B = (\Omega \setminus A) \cap B = (\Omega \cap B) \setminus (A \cap B) = B \setminus (A \cap B)$

So now we know that lemma is true, we can use it to conclude that:
$P(A^C \cap B) = P(B) - P(A \cap B)$

Looking at the right hand side, we have:
$=P(B) - P(A \cap B)$
$=P(B) - P(A)P(B)$ (using our key assumption)
$=(1-P(A))P(B)$
$=P(A^C)P(B)$

So we conclude that
$P(A^C \cap B) = P(A^C)P(B)$ as desired.

$QED$

3. $\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}$

4. Originally Posted by Plato
$\begin{array}{*{20}c}
{P\left( {A^c \cap B} \right)} & = & {P\left( B \right) {\color{red} - } P\left( {A \cap B} \right)} \\
{} & = & {P\left( B \right) {\color{red} - } P\left( A \right)P\left( B \right)} \\
{} & = & {P(B)\left( {1 - P(A)} \right)} \\
{} & = & {P(B)P\left( {A^c } \right)} \\

\end{array}$
Small typos fixed (in red).

A Karnaugh table is a visual way of making both the first line of Plato's proof and the conclusion of Last Singularity's lemma crystal clear.