It is much easier to solve this if you consider the probability of NOT all of the connections going down. To begin with, you find the probability of all the connections going down on one day, which is 0.001^k. Taking 1 minus that gives you the probability that there will be at least one connection up on a single day. The negation of having all connections down on at least one day of the year is having at least one connection up on every single day of the year, so you approach the problem that way.
