Suppose Cn is an ndimensional cube of edge length equal to 1

Suppose C(n) is an n-dimensional cube of edge length equal to 1. We say that a point p ? C(n) is near the boundary of the cube if it is not contained in the concentric cube having edge length of 1/2. Show that as n increases the probability that a randomly selected point in C(n) is near the boundary of the cube approaches 1.

Solution

C(n) is having n dimensions equal to 1 all sides.

The concentric cube inside will have all its sides at equal distance of 1/2 from the sides of the big cube C(n)

When n increases the cube sides almost become a circle losing its linear shape and becomes curved.

Thus the cuboid becomes a cylinder.

Hence in the boundary any point taken it will have equal distance from the concentric cube.

Thus the point approaching the boundary will be almost certain as n increases or in other words

Prob approaches 1.