I'm doing fine in all the other probability stuff I'm doing, but I got stuck in proving the following theorem:

Source:If $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ are any three events, then

$\displaystyle P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B\cap C)+P(A\cap B\cap C)$Probability and Statistical Inference, 7e, by Hogg and Tanis

Here is my work thus far:

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

This implies that $\displaystyle P(A\cup(B\cup C))=P(A)+P(B\cup C)-P(A\cap(B\cup C))$

Now, $\displaystyle P(B\cup C)=P(B)+P(C)-P(B\cap C)$

So, substituting this into the equation I get:

$\displaystyle P(A\cup(B\cup C))=P(A)+P(B)+P(C)-P(B\cap C)-{\color{red}P(A\cap(B\cup C))}$

I have no clue how to break up the part in red. Instead of working it out for me, give me a hint or two to help me get started.

Thanks in advance for any help!!

--Chris