Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 2 / 9E / b / i

Events $A$ and $B$ are independent whenFor a possible sum $2\le t\le 12$, let $N_t$ be the number of ordered die pairs with sum $t$. ThenIf $t-i\in \{1,\dots ,6\}$, then $\mathbb{P}(A_t\cap B_i)=1/36$; otherwise it is zero. Independence for a possible sum therefore requiresso $N_t=6$. This occurs only for $t=7$, and then $7-i$ is valid for every $i=1,\dots ,6$. ThusIf impossible sums are admitted as events of probability zero, they are trivially independent as well.