Pairwise independence vs independence 
Two fair dice are thrown. We have three events:

*

*A: The first die shows an odd number


*B: The second die shows an even number


*C: Both are odd or both are ven
Show that $A,B,C$ are piecewise independent but not independent.

My answer:
$P(A) = P(B) = P(C) = \frac{1}{2}$.
$P( A \cap B) = P( A \cap C) = P( B \cap C) = \frac{1}{4}.$
This means that all the events are pairwise independent. However:
$P(A \cap B \cap C) = 0$ while $P(A)P(B)P(C) = \frac{1}{8}$, so the events are not independent.
Is this correct (disregarding that I didn't explain how I got those probabilities)?

In this case you are using that $A$ and $B$ are independent iff $P(A\cap B)=P(A)·P(B)$
I think clarifying that is important. Beyond that, it's all correct.
They are pairwise independent because $P(A)·P(B)={1\over 4}$ too (and the same's true for $A$, $C$ and $B$,$C$).
The important part, though, is that you're correct.
