What is the probability that the sum of the squares of $3$ positive real numbers whose sum is less than $10$ is less than $16$?

Question

What is the probability that the sum of the squares of $3$ positive real numbers whose sum is less than $10$ is less than $16$?

This is how I understood the question:

Let $a,b,c\in\mathbb R^+$ with

$$a+b+c<10$$

Then find the probability such that

$$a^2+b^2+c^2<16$$

There are infinitely many positive real numbers. Know how to calculate probability?

I would like to draw a circle or triangle area. But I can't establish a connection with the triangle area or the circle.

The situations are also infinite. This question sounds as if it will be solved from the area of a figure. Do you think I am on the right track?

Nothing comes to mind.

Answer 1

Making use of geometric probability and Mathematica, one obtains

RegionMeasure[ ImplicitRegion[ x + y + z <= 10 && x^2 + y^2 + z^2 <= 16 && x >= 0 && y >= 0 && 
z >= 0, {x, y, z}]]/RegionMeasure[ImplicitRegion[ x + y + z <= 10 && x >= 0 && y >= 0 && z >= 0, 
{x, y, z}]]

$$\frac{8 \pi }{125} $$

Answer 2

Geometrically, the question is asking: Given a (uniformly-distributed?) random point in the tetrahedron bounded by the plane $z = 10 - x - y$ and the planes formed by the three axes, what is the probability that it lies within (1/8 of) a sphere of radius 4 centered at the origin. Divide the volume of the sphere segment ($\frac{32\pi}{3}$) by the volume of the tetrahedron ($\frac{500}{3}$).

Or, if geometry is not your forte, try a Monte Carlo simulation:

import random

def random_point():
    while True:
        # Generate random point in cube 0 <= x, y, z < 10
        x = random.uniform(0, 10)
        y = random.uniform(0, 10)
        z = random.uniform(0, 10)
        # Is it within the given tetrahedral region?
        if x + y + z < 10:
            return (x, y, z)

def estimate_probability(iteration_count):
    points_in_sphere = 0
    for i in range(iteration_count):
        x, y, z = random_point()
        if x ** 2 + y ** 2 + z ** 2 < 16:
            points_in_sphere += 1
    return points_in_sphere / iteration_count

The estimate_probability function returns an average result around 0.201. For more accuracy, use more iterations.