Let $X$ be a set containing n elements. Two subsets $A$ and $B$ of $X$ are chosen at random, the probability that $A\cup B=X$ is

Therefore, there are $4n$ ways in which we can randomly choose two subsets $A$ and $B$ of $X$.

Now we will count the number of ways in which $A\cup B=X$

In this case, we want every number $i\in X$ which satisfy one of the three
There are $n$ numbers in set $X$. 

Therefore, there are $3^n$ ways in which we can randomly choose two subsets $A$ and $B$ of $X$ such that $A\cup B=X$.

So, the probability that two randomly chosen subsets $A$ and $B$ of $X$ satisfy $A\cup B=X$ is equal to $\frac{3^n}{4^n}={\left(\frac{3}{4}\right)}^n$.