Approximate the proportion of integers with neighboring factors
Jelly,  14 13  10 bytes
-1 using Erik the Outgolfer's idea to take the mean of a list of zeros and ones.
-3 by using 3-indexing (as allowed in the question) - thanks to Dennis for pointing this out.
ḊPƝḍⱮ!Ẹ€Æm

A monadic Link accepting an integer, n+2, which yields a float.
Try it online! (very inefficient since it tests divisibility over the range \$[2, (n+2)!]\$)
(Started out as +2µḊPƝḍⱮ!§T,$Ẉ, taking n and yielding [numerator, denominator], unreduced)
How?
ḊPƝḍⱮ!Ẹ€Æm - Link: integer, x=n+2
Ḋ          - dequeue (implicit range of) x  - i.e. [2,3,4,...,n+2]
  Ɲ        - apply to neighbours:
 P         -   product                             [2×3,3×4,...,(n+1)×(n+2)]
     !     - factorial of x                        x!
    Ɱ      - map across (implicit range of) x! with:
   ḍ       -   divides?                            [[2×3ḍ1,3×4ḍ1,...,(n+1)×(n+2)ḍ1],[2×3ḍ2,3×4ḍ2,...,(n+1)×(n+2)ḍ2],...,[2×3ḍ(x!),3×4ḍ(x!),...,(n+1)×(n+2)ḍ(x!)]]
       €   - for each:
      Ẹ    -   any?  (1 if divisible by any of the neighbour products else 0)
        Æm - mean