We are given a sphere that fits inside a cube. The sphere touches each side of the cube. Let the radius of the sphere be r.
We will find the probability that a point chosen at random inside the cube is also inside the sphere. The desired probability is a geometric probability, so we need to find the volume of the sphere and the volume of the cube. Since the radius of the sphere is r, and since the sphere touches the each side of the cube, one side length of the cube is 2r — the diameter of the sphere.
Volume
Sphere
4/3π r^3
Cube
(2r)^3= 8r^3
Now we will find the probability that a randomly chosen point from the volume of the cube is also within the volume of the sphere.
P(Point in sphere)=Volume of the Sphere/Volume of the Cube
