Probability a Group Sits Together at a Circular Table

Fix a positive integer $n$. There are $3n$ people who sit around a circular table with $3n$ seats, uniformly at random. The $3n$ people are divided into 3 groups of $n$ people each. Find the probability that **at least one** of the three groups has all $n$ of its members seated adjacent to one another. Rotations of a seating arrangement are considered identical. Report the answer for $n = 5$, rounded to the nearest ten-thousandth.