I do not see how the probability can be larger than 1/4. Here is my reasoning: Let the machines be numbered 1,2,...,n, and let the lights be a and b, so that 1a2b means connecting the "a" bulb of machine 1 to the "b" bulb of machine 2. I am assuming that the bulbs light up randomly, with equal probability of bulb "a" and bulb "b" lighting up on each machine, and that choice of which bulb to light up is independent on each machine.

Now any connection between the two bulbs on the same machine, like 1a1b, 2a2b, ... nanb will NEVER have both bulbs light up at the same time, because each machine chooses one bulb randomly to light up. Any connection between two different machines will succeed exactly 1/4 of the time, because the two bulbs light up independently with probability 1/2 each.

Therefore, the best strategy is to avoid connecting the two bulbs on the same machine, and only connect bulbs on different machines. Since all the connections between different machines have an equal probability of success, it doesn't matter how you connect them or how many connections you have.

The smallest number of machines where you can do this is n=2, and the simplest connections is to just make one connection 1a2a (or 1a2b, or 1b2a, or 1b2b, it doesn't matter). Then on average, the number of successes will be 1/4 of the number of random trials.

Perhaps I am misunderstanding the problem. Should I not assume the machines are independent? When you say "connect the bulbs" do you mean just to form a mathematical set of two bulbs, just to see if they both light up? Or do you mean some kind of physical connection that would change the probability of lighting up?

--Stuart Anderson