Mutual independence when rolling a sequence of dice

Question

Suppose we are rolling say 5 symmetrical dice, one after another, and we assume uniform distribution of probability, i.e. for each event $u$ of the form $u = (a, b, c, d, e)$, where $a,b,c,d,e$ are numbers from $1$ to $6$, we let $P(u) = 1/6^5$. Is it then true that the collection of every set of every possible die roll at every place in the sequence is mutually independent? In other words, is it true that the set $\{A_{11}, A_{12}, \dots, A_{55}, A_{56}\}$ is mutually independent, where $A_{ij} = \{j\;\text{pips on roll number}\;i\}$?

Answer 1

No, because $\Pr(A_{ij}\cap A_{ik})=0\neq\Pr(A_{ij})\Pr(A_{ik})$ whenever $j\neq k$. Any set which does not include any pair of this form is mutually independent: the roll on each die is independent of anything that happens on any combination of the other dice.