Suppose that time is divided in discrete steps. There are n people and a single shared computer than can only be accessed by one person in a single step: if two or more people attempt to access the computer at the same step, then everybody is "locked out" during that step. At every step, each of the n people attempts to access the computer with probability p.

(a) (5 points) Determine the probability that a fixed person i succeeds in accessing the computer during a specific step. (b) (5 points) How would you set p to maximize the above probability? (c) (10 points) For the choice of p in part (b), upper bound the probability that person i did not succeed to access the computer in any of the first t = en steps. Hint: Use inequality 1 in Remark 1. (d) (10 points) What is the number of steps t required so that the probability that person i did not succeed to access the computer in any of the first t steps is upper bounded by an inverse polynomial in n? (e) (15 points) How many steps are required to guarantee that all people succeeded to access the computer with probability at least 1 1/n (that is, with high probability)? Hint: You may want to first upper bound the probability of the complementary event.