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)?

Yes, this is correct. The definitions are as follow (see here).
A finite set of events $\{A_i\}$ is pairwise independent if and only if every pair of events is independent — that is, if and only if for all distinct pairs of indices $m, k$,
$$
\mathrm{P}(A_m \cap A_k) = \mathrm{P}(A_m)\mathrm{P}(A_k).
$$
A finite set of events is mutually independent if and only if every event is independent of any intersection of the other events — that is, if and only if for every $n$-element subset ${A_i}$,
$$
\mathrm{P}\left(\bigcap_{i=1}^n A_i\right)=\prod_{i=1}^n \mathrm{P}(A_i).
$$
You have shown that the events $A,B$ and $C$ are pairwise independent,but not mutually independent.
