The sphere was continuously forced to change its rotational axis as the direction of the magnetic forces were changed by the machine, and thus a quick, random spin and tumble was gained.

He then instructed three crackshot marksmen to aim, shoot at and hit the sphere, each one at five second intervals. The marksmen were each using a bullet from a specially coded set of three that allowed the mathematician to distinguish the three hits of any one round from the hits of other rounds. The marksmen each fired at the target 100 times.

After the target was hit 300 times, the mathematician removed the sphere from the machine and identified the hits from each round of three. Then he drew his conclusion.

What is the probability that the three points of impact from each round are on the same hemisphere?