A regular polygon with $100$ sides is inscribed in a circle. What is the probability that three randomly chosen vertices of this polygon form a right angled triangle?
An inscribed triangle is a right triangle if its largest side is a diameter of the circle. If the vertices of the regular $100$-gon are numbered $0$ through $99$ then a diameter connects vertices whose numbers differ by $50$. Without loss of generality we may choose to label the first vertex chosen as $0$. Given two numbers $x$ and $y$ selected uniformly, randomly, without replacement from $1$-$99$ we seek the probability that either number is $50$, or that $\lvert x-y\rvert = 50$. Note that these are disjoint events they cannot both be true.
I will end at this point in case this is an unlabeled homework assignment. The rest is rather straightforward.