A device with two light bulbs. Suppose you are allowed to combine N such devices. Everytime when a switch is pressed, each device has one of these two light bulb turn on randomly. You are required to make a connection between any pair of light bulbs among the N devices such that for every 3 connections that you made, there must be at least one connection that will link up two light bulbs turning on. Suppose the switch is pressed to give random lighting of bulbs and the process will go on indefinitely, what is the minimum number of such devices and connections (with the way of connections shown as diagram) that will have the highest probability of achieving 1/3 of your total connections that will link up two light bulbs turning on?
I may not frame the question properly but hope you can understand its content. Can anyone interested in this question give me the answers and a detailed explanation (or generalized this specific problem)? Thank you in advance.