1."Y=AX/((AX)+(B(1-X))" How have you got this formula? Sorry, my probability math is a bit rusty, so maybe I'm missing something obvious.

2.I find it a poor choice to use two islands as your example. In context of this problem we deals with two sets, and each set >1. Even more, such example biases us to think that an observer to be moke likely to belong to a set of worlds where catastrophe hasn't happened, as there are only two islands. It doesn't need to be the case. While each world that experienced catastrophe is less populated, combined population of post-catastrophe worlds can be still greater than population of no-catastrophe worlds if there is too many of post-catastrophe worlds and too few no-catastrophe worlds. IMHO, it would be better to use set A (islands where volcanic eruption happened) and set B (islands where were no volcanic eruptions). 

3."So the probability of being on an island that has had a volcanic eruption is 1/3 " Why would we want to know this? I think that calculation of risk of catastrophe (i.e. share of ruined worlds/islands) is much more relevant.  

4.It's unclear to me how you select worlds among which M.A.D. either happened or didn't. For an example, in my comment I limited myself to all worlds where I was born. If you don't do the same, then you will run into problems. Consider this. There are parallel worlds where Roman civilization never crumbled. Where they had time to achieve everything that current Western civilization achieved + several addtional centuries to go beyond this. In 2022 there would be Roman worlds that colonized several planets of Solar system, maybe even terraformed them and found ways to sustainably support much bigger population than 8 billions. It seems plausible for population to be distributed heavely in favor of Roman worlds in current year 2022. Yet you and me aren't is a Roman world. Curious, don't you think?

I don't know how much my calculations are different from yours as I hasn't been able to comprehend how to use your formula. Can you give me an elaborate example of using it, step-by-step? 

"We can't know that it is more likely that we are in a world that hasn't experienced nuclear war, we are however justified in believing that it is more likely that we would be in a world that hasn't experienced nuclear war." Sorry, but I fail to see the difference between "it is more likely that we are in a world that hasn't experienced nuclear war" and "it is more likely that we would be in a world that hasn't experienced nuclear war"

I think I just found a way to roughly estimate nuclear risk by using assumptions about existence of parallel worlds and that I exist in the world that hasn't ever experienced nuclear war because this is more probable to be in such world. x is total number of worlds whare I was born. Let's also assume that time flows at the same rate in all worlds, so in each world current year is 2022.  In some worlds I, for various reasons, died. In other ones I'm still alive. Obviously, I'm more likely to be alive in worlds where there are more people. And worlds where nuclear war happened will be significantly less populated. I read that nuclear war in our world would kill 5 billions of people, out fo current 8 billion people. So, in the most generous case, post-nuclear-war world would have about 3 billions people, while  worlds that avoided nuclear war would have about 8 billions of people. Now we can make inequality, x*y*3<x*(1-y)*8. y means risk of nuclear war, namely share of worlds that experienced it since my birth up to now. Total population of all post-nuclear-war worlds combined must be less than total combined population of all no-nuclear-war-worlds, as I judge that my current existence in no-nuclear-war world is evidence that there is higher chance for me to exist in no-nuclear-war world in this year. We can divide both sideds by x to get rid of it, and then solve for y. https://www.wolframalpha.com/input?i=y*3%3C%281-y%29*8   As you can see, y<8/11, or in other words, no more than 73% of all worlds (where I was born) got nuked since my birth if we assume that this inequality is true.