Draw a diagonal of the square to divide into 2 identical right angled triangles.
Each triangle has a base equal to diameter of the circle, d (or 2r) , and a height equal to its radius , r
Area of square = area of 2 triangles
= 2 x ½ x 2r x r
= 2r²
Area of circle = πr²
Remaining area not occupied by the square = πr² - 2r² = (π-2)r²
The shaded part is equal to the other 3 parts that are not occupied by the square.
Area of shaded part = ¼ of remaining area = ¼(π-2)r²
Probability required = area of shaded part / area of circle
= ¼(π-2)r² / πr²
= ¼(π-2) / π
= ¼ - 1/(2π)