It is likely you may get some brain teasers in your interviews. The followings are three questions, each of which represents the format of those brain teasers. Show the details of derivations and calculations for the following problems. (a) There is a circle with radius R. Randomly pick three points on the circle (each point on the circle has an equal chance to be picked) and connect them by lines to form a triangle. What is the probability that such a triangle is an acute triangle? (b) There is a rope. Randomly pick two different points on the rope (each point on the rope has an equal chance to be picked). Cut the rope into three segments according to these two points. What is the probability that these three segments can form a triangle? (c) Toss a fair coin for infinity times. All tosses are independent. For each toss, if it is head, you will get 1 point. If it is tail, you will get 2 points. In the entire process, what is the probability of ever having n points