Step 1: Understand what the question asks.
Alu I is a restriction enzyme that recognises and cuts at a specific $4$ base pair sequence. We want how often, on average, this site shows up in a random genome, because that sets how often the enzyme cuts.
Step 2: Set up the probability for one position.
If bases are randomly distributed, each position is equally likely to be $A$, $T$, $G$, or $C$. So the chance that one particular base sits at a given spot is $\dfrac{1}{4}$.
Step 3: Extend to all four positions of the site.
The recognition site is a fixed run of $4$ specific bases. Each must match independently, so we multiply the per-position probabilities together.
Step 4: Do the multiplication.
\[ P = \left(\frac{1}{4}\right)^4 = \frac{1}{4^4} = \frac{1}{256} \]
Step 5: Interpret the number.
A probability of $\dfrac{1}{256}$ means the exact $4$ base sequence appears on average once in every $256$ base pairs, so the enzyme cuts about once per $256$ bp.
Step 6: Eliminate the distractors.
$\dfrac{1}{4}$ is just one base, $\dfrac{1}{2^4} = \dfrac{1}{16}$ wrongly uses $2$ choices per base, and $\dfrac{1}{1296} = \dfrac{1}{6^4}$ wrongly uses $6$ choices. The correct count uses $4$ bases, giving $4^4 = 256$.
\[ \boxed{\frac{1}{256}} \]