P.S. Im saying melody or asinus in the title because their the only moderators active. But anyone can help!

A point in space (x, y, z) is randomely selected so that -1 ≤ x ≤ 1, -1 ≤ y ≤ 1, -1 ≤ z ≤ 1. What is the probability that \(x^2+y^2+z^2\le 1\)

supremecheetah May 27, 2023

#1**0 **

The given space is a cube with sides of length 2 centered at the origin. The condition x2+y2+z2≤1 represents a sphere with a radius of 1 centered at the origin. The two regions intersect in a four-sided pyramid with vertices at (1, 1, 1), (1, -1, -1), (-1, -1, 1), and (-1, 1, -1). The volume of the cube is 8, and the volume of the pyramid is 1/3 * 2 * 2 * 2 = 8/3. Therefore, the probability that a randomly selected point in space satisfies the condition x2+y2+z2≤1 is 8/3 / 8 = 1/3.

Guest May 27, 2023