1. ## probability proof

Let A, B, and C be any three events. Show that:
(1) $P(A)=P(B)$ if and only if $P(A \cap B^c)=P(A^c \cap B)$
(2) Given $P(A)=0.5$ and $P(A \cup (B^c \cap C^c)^c)=0.8$, determine $P(A^c \cap (B \cup C))$

For first part, I don't know how to start at all. Plus, with "if and only if", does that mean I need to prove from two directions?
For second part, I did draw Venn Diagram, but still don't feel that is very helpful, except I did find $P(A \cup B \cup C)=0.8$, am I right? What should I do?
Thanks a lot.

2. Originally Posted by tsang
Let A, B, and C be any three events. Show that:
(1) $P(A)=P(B)$ if and only if $P(A \cap B^c)=P(A^c \cap B)$
For any two events this is true:
$P(A)=P(A\cap B)+(A\cap B^c)$ and
$P(B)=P(B\cap A)+(B\cap A^c).$

Use that to prove:
1) If $P(A)=P(B)$ then $P(A \cap B^c)=P(A^c \cap B).$
and
2) If $P(A \cap B^c)=P(A^c \cap B)$ then $P(A)=P(B)$.

3. Originally Posted by tsang
(2) Given $P(A)=0.5$ and $P(A \cup (B^c \cap C^c)^c)=0.8$, determine $P(A^c \cap (B \cup C))$
For second part, I did draw Venn Diagram, but still don't feel that is very helpful, except I did find $P(A \cup B \cup C)=0.8$, am I right?.
That is correct. Now find the area shaded in yellow.

4. Thanks a lot Plato. I need to keep trying on part 1, it is still a bit hard for me. If I still can't get it, I may ask you for help again. Thank you.

5. Originally Posted by tsang
If I still can't get it, I may ask you for help again. Thank you.
You cannot apply $P(B\cap A)+(B\cap A^c)=P(A\cap B)+(A\cap B^c)$ and subtract?
What gives here?

6. Originally Posted by Plato
You cannot apply $P(B\cap A)+(B\cap A^c)=P(A\cap B)+(A\cap B^c)$ and subtract?
What gives here?
Thank you very much, Plato, sorry I was just thinking too much, because I did too much Real Analysis last year, so when I saw that "if and only if" statement, I got scared and thought I need to write heaps of detailed proof. So yes, you are absolutely right, it was just me thinking too much.

May I ask about part 2 again, I calculated the result is just 0.3, in fact can just use 0.8-0.5=0.3, but the only thing is could you please give me advice on how to write about it? Because I feel only use Venn diagram to show is not quite mathematical, could you please tell me if there is anyway to write about it? I did get A U B U C=0.8, but that is really because I worked out from the diagram. Do you think I can do it algebracally? Thanks again.