Let $A_{1}, A_{2}, \ldots, A_{n}(n \geqslant 2)$ be events in a sample space. For each of the following statements, either prove the statement or provide a counterexample.

(i)

$$P\left(\bigcap_{k=2}^{n} A_{k} \mid A_{1}\right)=\prod_{k=2}^{n} P\left(A_{k} \mid \bigcap_{r=1}^{k-1} A_{r}\right), \quad \text { provided } P\left(\bigcap_{k=1}^{n-1} A_{k}\right)>0$$

(ii)

$$\text { If } \sum_{k=1}^{n} P\left(A_{k}\right)>n-1 \text { then } P\left(\bigcap_{k=1}^{n} A_{k}\right)>0$$

(iii)

$$\text { If } \sum_{i<j} P\left(A_{i} \cap A_{j}\right)>\left(\begin{array}{c} n \\ 2 \end{array}\right)-1 \text { then } P\left(\bigcap_{k=1}^{n} A_{k}\right)>0 \text {. }$$

(iv) If $B$ is an event and if, for each $k,\left\{B, A_{k}\right\}$ is a pair of independent events, then $\left\{B, \cup_{k=1}^{n} A_{k}\right\}$ is also a pair of independent events.