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