A board has 16 squares as shown in the figure:

Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is:

Total ways to choose 2 squares: $$\binom{16}{2} = \frac{16 \times 15}{2} = 120$$.

Number of pairs with a common side (Adjacent pairs):

Horizontal adjacencies: In each of the 4 rows, there are 3 pairs $$\implies 4 \times 3 = 12$$.

Vertical adjacencies: In each of the 4 columns, there are 3 pairs $$\implies 4 \times 3 = 12$$.

Total common side pairs = $$12 + 12 = 24$$.

Pairs with NO common side: $$120 - 24 = 96$$.

Probability: $$P = \frac{96}{120} = \frac{4}{5}$$.