1. Probability that all selected students are girls = Ways of selecting 4 girls out of 5 girls/ Total number of ways of selecting 4 students out of 8 total students.
2. The vertices of the inscribed squares bisect the sides of the larger squares. Using, this we can find out that the area of the inscribed squares is also halved with respect to the larger squares. This can be proven, if we assume that the side of the largest square is of 2 cm, then, half side length is 1 cm and hence, the side of the inscribed square (using pythagoras theorem in any of the 4 triangles so formed) will be sqrt(2) cm. So, the area of the inscribed square is 2 cm2, while that of the original larger squares was 4 cm2 (the area of a square is side*side). So, the inscribed squares have half the area as compared ot the square in which they are inscribed.
In this figure, there are 4 inscribed squares within the largest one and the area of each one of those is half of the area of the square in which they are inscribed. This means that the area of the smallest innermost square is 1/(2^4) times the area of largest square i.e. the area of the smallest square is 1/16th the area of the largest square.
Here, when we talk of the shaded region, it is between the innermost smallest square and the square within which it is transcribed. Since, the area of the smallest square is half of the area of the square withinwhich it is transcribed, the remaining portion of that square (which is not captured within the smallest innermost square) is also half the area of the larger circumscribing square. Furthermore, in this figure only 2 corners of the remaining area are shaded, so the area shaded is half of the area of the smallest square even.
Therefore,1/2 *1/16 = 1/32 is the fraction of the shaded area to the largest square.