This problem is based on a Citadel Securities interview problem described here.
Eight independent switches have probabilities 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 4/5, and 2/5 of being on. An integer C is chosen uniformly at random from 0 through 8, independently of the switches. What is the probability that at least C switches are on?