Let $\mathcal{R}$ be the set of all binary relations on the set $\{1,2,3\}$. Suppose a relation is chosen from $\mathcal{R}$ at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is ________.

Let $A = \{1, 2, 3\}$. The number of elements in $A$ is $n = 3$.

1. The total number of pairs in $A \times A$ is $n^2 = 3^2 = 9$.

2. The total number of binary relations on $A$ is:

   $$|\mathcal{R}| = 2^{n^2} = 2^9 = 512$$

3. For a relation to be reflexive, it must contain all $n = 3$ diagonal pairs $(1, 1), (2, 2), (3, 3)$. The remaining $n^2 - n = 9 - 3 = 6$ non-diagonal pairs can either be present or absent independently.

4. Thus, the number of reflexive relations is:

   $$2^{n(n-1)} = 2^{9-3} = 2^6 = 64$$

5. The probability of choosing a reflexive relation at random is:

   $$P = \frac{2^6}{2^9} = \frac{1}{2^3} = \frac{1}{8} = 0.125$$

Rounded to 3 decimal places, the answer is 0.125.