Take an array of y random integers in the inclusive range 0 to 99, each integer written in the form ab, where a is the first digit in the integer and b the second. For example, integer 9 is written as 09 for this purpose, a=0 and b=9.
Two of the array members a(1)b(1) and a(2)b(2) and are linkable if a(1)=b(2), or b(1)=a(2).
Let us create a looped chain of linkable array members of length x. An example of a looped chain of length 4 is 09,95,57,70. 70 links back to 09.
Additional rules: (i) in the random generation of array members, repeated integers have a separate member identity in the array (ii) array members are permitted to be members of multiple looped chains, but only to appear once in each chain
Then:
(I) For the defined array, what is the probability that there exists one or more looped chain(s) of length x, with x < y?
(II) Is there a method to generate, or a formula for generating, estimates of how many different possible chains of length x exist within an array of length y?
(III) Is there a formula/ method to generalise this in any number base?
Denote the base by $q$.
I'll assume that by “random integers” you mean that $y$ two-digit integers are independently uniformly chosen in the range $[0,q^2-1]$.
Question $1$ seems intractable without a computer. However, for $y\ll q$, the probability of multiple looped chains is small, and the probability of obtaining at least one looped chain is approximately the expected number of looped chains from Questions $2$ and $3$.
Questions $2$ and $3$ are straightforward to answer.
There are $\frac{y!}{(y-x)!}$ potential looped chains of length $x$. Each of them has probability $q^{-x}$ of actually being a looped chain. Thus the expected number of looped chains of length $x$ is $\frac{y!}{(y-x)!}q^{-x}$, and the expected total number of looped chains is
$$ \sum_{x=2}^y\frac{y!}{(y-x)!}q^{-x}=\frac{y!}{q^y}\sum_{k=0}^{y-2}\frac{q^k}{k!}\approx\frac{y!}{q^y}\sum_{k=0}^\infty\frac{q^k}{k!}=\frac{y!\mathrm e^q}{q^y}\;. $$
The approximation is valid for $y\gg q$.