Introduction:
In this problem, we are given a regular hexagon and we need to find the probability that the triangle formed by choosing three of its vertices at random is an equilateral triangle.

Approach:
To solve this problem, we can first count the number of equilateral triangles that can be formed from the six vertices of the hexagon. Then, we can calculate the total number of possible triangles that can be formed from these vertices. Finally, we can divide the count of equilateral triangles by the count of total triangles to find the probability.

Counting Equilateral Triangles:
To count the number of equilateral triangles, we can start by selecting any vertex of the hexagon. Once the first vertex is chosen, there are two possible choices for the second vertex, as it must be adjacent to the first vertex. Finally, for the third vertex, there is only one choice, as it must be adjacent to both the first and second vertices. Therefore, there are 2 equilateral triangles that can be formed for each vertex of the hexagon.

Since there are 6 vertices in total, the total count of equilateral triangles is 6 * 2 = 12.

Counting Total Triangles:
To count the total number of triangles, we can use the combination formula. Since we need to choose 3 vertices out of 6, the total count of triangles is given by C(6, 3) = 6! / (3! * (6 - 3)!) = 20.

Calculating Probability:
The probability of choosing an equilateral triangle is given by the count of equilateral triangles divided by the count of total triangles. Therefore, the probability is 12 / 20 = 3 / 5.

Conclusion:
The probability that the triangle formed by choosing three vertices of a regular hexagon at random is an equilateral triangle is 3/5, which can be simplified to 1/10. Therefore, option A is the correct answer.