In a random arrangement of the letters of the word 'UNIVERSITY',...

The total number of arrangements of the letters of the word 'University' is $\frac{10!}{2!}$ as there are two i's. Regarding 2I's as one letter, number of ways of arrangement in which both I's are together $=9$ ! So, the number of ways in which 2I's are not together = $\frac{10!}{2!}-9$ ! Hence, the required probability $$\begin{aligned} & =\frac{\frac{10!}{2!}-9!}{\frac{10!}{2!}} \\ & =1-\frac{2!9!}{10!}=\frac{4}{5} .\end{aligned}$$