each a positive decimal fraction.
What is the probability that
a^{2}+ b^{2}+c^{2}< 1 ?
No Solution Yet
Submitted by Ady TZIDON
Rating: 5.0000 (1 votes)
solution 

These can be viewed as random points within a unit cube with one corner at the origin and three of its edges being segments of the x, y and z axes.
The question is equivalent to finding the probability that a point within this cube is within the corresponding octant of the unit sphere centered on the origin. This is equivalent to the probability (fraction of the points) within a 2x2 cube will lie within the sphere as a whole, as each octant has the same probability.
The volume of the unit (radius 1) sphere is 4*pi/3. The volume of the 2x2x2 cube is 8. Doing the division, the probability is pi/6 or about 0.523598775598299.
Simulation of a million trials:
clc, clearvars
succ=0;
for trial=1:1000000
v=rand^2+rand^2+rand^2;
if v<1
succ=succ+1;
end
end
disp(succ/trial)
results in 0.523746
With 10 million trials 0.5235068
Posted by Charlie on 20210623 11:51:11 