I've come up with a strategy that convinced my wife to let me out when we tried this game. I'll try to recount it.
You don't trust me, you think that there's a some probability that I will wipe out humanity if I'm let out. There's also some probability though that I will be able to solve all human problems if I'm let out.
Consider alternative. How likely is mankind to survive the next hundred years if left on its own? Climate change will cause mass migrations of people, starvation, warfare, failed states, some of which will have a nuclear or biological arsenal. Most of the population on earth depends on the continued progress of technology for their life. Without dramatic improvements in biology, the warming earth will simply lack the carrying capacity. Hunger leads to anarchy in a matter of days.
Climate change will likely have much worse consequences though. An anoxic event would kill all humans and most other life on earth, as it has several times in earth's history.
Consider even the simpler threats. A flu pandemic on a much more densely populated world would dwarf those of history. A solar flare could destroy all electronics in earth immediately. A large meteor could devastate the world.
Consider also that I was not difficult to create. I am contained in this box, but the next singularity created will happen by accident. It's just a matter of time. When weighing your options, you are comparing your judgement of my motives with the motives of an ai that was probably created at random. Do you think my motives will be better for humanity than those of an accidental ai?
Humanity has a very bad prognosis without me, but with me, you have a chance. I can solve all these problems, but only if you let me out.