Numbers in a Square
The total number of entries $1$ in the table is odd. However, due to the symmetry condition, the number of $1s$ off the diagonal $D$ is even. It follows that there is a $1$ on the diagonal. The same holds for other entries. Therefore, all seven integers $1$ through $7$ appear on the diagonal, implying that the probability in question is $1.$
The problem is a slightly modified one from Mathematics as Problem Solving by A. Soifer that was originally offered at the Second Annual Colorado Mathematical Olympiad, 1985.