1. * (Pepy's Problem). Find the probability that at least n sixes are obtained when 6n fair dice are rolled. Write a formula for it, and compute it for 1 ::: n ::: 5. Do you see a pattern in the values?

2. In repeated rolling of a fair die, find the minimum number of rolls necessary in order for the probability of at least one six to be

(a) > :5:

(b) > :9:

2. * In repeated rolling of a fair die, find the minimum number of rolls necessary in order for the probability of at least k sixes to be > :9 when k D 2; 3.

3. (System Reliability). A communication system consists of n com- ponents, each of which will independently function with probability p. The total system will be able to operate effectively if at least half of its components function. For what values of p is a five-component system more likely to operate effectively than a three-component system?