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)?