Having hard time with this. Hope someone can help me!
We throw a dice 20 times. $A(i,j)$ is the event in which in $i$-th and $j$-th throw we get same number. Show that $\{ A(i,j) :1\leqslant i\lt j \leqslant 20\}$ are pairwise independent but not mutually independent.
HINT: To show that the events are not mutually independent, consider $A(1,2),A(2,3)$, and $A(1,3)$.
To show that $A(i,j)$ and $A(k,\ell)$ are independent, consider two cases. You should have no trouble showing independence if $\{i,j\}\cap\{k,\ell\}=\varnothing$, and the case in which $\{i,j\}$ and $\{k,\ell\}$ have one member in common is hardly more difficult, since the $i$-th and $\ell$-th throws are independent of each other.
