Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 8 / c / v

First restrict to integers in the marginal supports: $2\le i\le 12$ and $-5\le j\le 5$. Their marginal probabilities are positive. By part (i), a feasible joint pair identifies a unique ordered pair of rolls, so its joint probability is $1/36$; an infeasible pair has joint probability zero.

Put $a_i=6-|i-7|$ and $b_j=6-|j|$. If the two events were independent, a feasible pair would requireSince $1\le a_i,b_j\le 6$, equality forces $a_i=b_j=6$, hence $(i,j)=(7,0)$. But that pair is infeasible. An infeasible pair cannot give independence either, since its joint probability is zero while the product of its positive marginal probabilities is positive. ThusIf all real values of $i,j$ are allowed literally, an impossible event is independent of every event. The complete answer without a support restriction is therefore