Question

6. Finding a Healthy Subring A group of n people po, ..., Pn-1 are standing around in a circle holding hands. When we talk about these people, we take indices modulo n so that, for example Pn+5 = 25.s of these people are sick, the rest are healthy. 1. Let k < 100 be an integer. Pick a uniformly random ie {0,...,n - 1} and let X denote the number of sick people among the k people pi, pi+1, ... , Pi+k-1. What is E(X)? 2. Argue that if we have n = 70 people standing around in a circle and s = 39 of them are sick, then there always exists 7 consecutive people pi, ... , Pi+6 at least 4 of whom are healthy. (This should be a bit surprising, since 4/7 > 1/2 of these people are healthy but 39/70 > 1/2 of all people are sick.)

Answer 1