| medium | probability |
A very sharp, consistently skillful blind archer aimed for the center of a circular board and shot 2 arrows. He is expected to hit the aim, but doesn't hit it for sure. The archer is told that his first shot is better than second. He tried one more shot. What is the probability that this 3rd shot is the best shot among 3? (ie, Probability that 3rd arrow lands closer to center than his first two shots?)
Enumeration; Symmetry between random variables.
1/3
Suppose x1, x2 and x3 are the distances of the arrows from center. As the archer is consistent, we can use symmetry, i.e., there are six equally likely cases: x1<x2<x3, (or 5 others, which are its permutations). Since archer is told that x1<x2, we are left with following equally likely cases: x1<x2<x3 , x2<x1<x3 , x1<x3<x2 Among these three, one is favorable. Hence The probability of last shot being best is 1/3.
Notice that if there were (N-1) arrows, with first being better than rest, and then he shoots Nth arrow. The probability that Nth shot is best is 1/N.