# Spinning Spherical Target: Probability for Three Marksmen to Hit on the Same Hemisphere

Date of Exam:

**Problem**

Three marksman simultaneously shoot and hit a rapidly spinning spherical target. What is the probability that the three points of impact lie on the same hemisphere?

A. 0 | C. 1 |

B. 1/2 | D. 2/3 |

**Answer Key**

[ C ]

**Solution**

A great circle can always be drawn through any two distinct points on the surface of the sphere.

Say the marksman will hit the points *P*_{1}, *P*_{2}, and *P*_{3}. Draw a great circle through *P*_{1} and *P*_{2} dividing the sphere into two hemispheres. Since *P*_{1} and *P*_{2} are on the boundary of these hemispheres, they belong to both hemispheres. Now *P*_{3} can be on any of the two hemispheres. Hence, the three points will always be on the same hemisphere.

Probability, *P* = 1.0 answer [ C ]