Systemic Stability dressed in Statistical Sensationalism

While playing Fallout 3, I found myself marveling once again that we have not totally annihilated ourselves as a species. In fact, I am still amazed that nothing came of the cold war. The fact that a nuclear attack on any country would result in an equally devastating counterstrike seems like an easily dismissed concern in the eyes of a psychopath with determination, no sense of self preservation, and the right connections - or so mass fiction would seem to hint.

But it hasn't happened yet. I live in what is undoubtedly the most hated country in the world, and the largest aggression-based atrocity we have had to weather lately involved a four digit death toll at the receiving end of some planes, or possibly some explosives, because those steel beams in the wreckage looked shopped and I've seen quite a few pixels in my days.

A coworker presented me with an interesting probabilistic thought experiment in the field of reliability. Let's say you have a somewhat complex system of 100 parts, and any given part is 95% reliable (it will work appropriately 95% of the time and fail the other 5%). Let's say this is a touchy system; if one component fails, then the entire system fails. The probability of the system working is 0.95 raised to the 100th power, which is 0.00592, a little over half of a percent. There is over a 99% chance that such a system would fail. Though each part is almost completely reliable, the fact that each part is necessary toward our goal of success makes the system holistically unreliable. If we can increase the reliability of every component to 99%, there is only about a 37% chance the system would work. We need to ensure about a 99.3% reliability rate on each component to even get the odds of a coin flip.

There are approximately 6.7 billion people on this planet. If we consider each person to be a component of the system we call humanity (or if we want to be a little less egocentric, the system we call planet Earth), I wonder what we can estimate our component reliability to be. If there was "only" a 5% chance, per person, of that person launching a nuke and instigating our final hours (95% chance against), the odds of us surviving another day are too low for my windows calculator to display. The same can be said if the odds of pacifism are 99% per person. Same for 99.99%.

Assuming we can't have 100% odds against a nuclear holocaust, what percentage would you feel comfortable with? Would you feel comfortable if the odds of such an Armageddon happening tomorrow were only 1%? Let's see what level of component reliability we have to have to obtain that goal. If the odds of every human being behaving and not finding some way to trick a few nations into playing fallout volleyball was 99.999999% per person, would you feel safe? I wouldn't, as 99.999999% raised to the 6.7 billionth power is 7.98e-30, or 1 in 125,236,359,038,394,908,283,678,232,890. Playing poker with a new hand played every minute, you're just as likely to be dealt a royal flush each hand for 366,720,740,041,699,000 years (about 26 million times the established age of the universe) as you are to avoid witnessing a nuclear winter tomorrow if that's the only per-person reliability we can expect.

Tacking on another couple of 9's, giving us a per-person reliability of 99.99999999%, puts us at the halfway mark. Heads and you can sigh in relief, tails and you should be signing up for lodging in a vault, preparing to drink gallons of Nuka Cola to replenish health with relatively low radiation poisoning, and practicing your melee skills since post-Armageddon low-levels have to be so close to raiders and mutants to use their guns effectively that they just as well utilize the extra damage that a readily available sledge hammer offers. Also, only music from the 1930's will survive the explosions, and don't expect anyone to find time over the next two centuries to write new songs.

If we want to reach our original goal of a 99% chance of avoiding a nuclear war tomorrow, we have to up the odds in our favor slightly more - to about a 99.99999999985% pacifism rate per person. That's quite a high component reliability that we must maintain for our system to work isn't it?

But wait, there's more!

The above figures will only get us through the next day, remember? I don't know about you, but I would like the world as we know it to last a little bit longer, at least another year. If we assume that a 99.99999999985% pacifism rate per person guarantees us a 99% chance of surviving for the next 24 hours, we can calculate that the odds of surviving for another week are about 93.2% (99% raised to the 7th power). Still pretty favorable, though perhaps not as much as we would like as we are talking about the largest death toll of our species in like, ever. But raise that 99% to the 365th power (366th if it's a leap year, but we'll aim a little lower for now) and the odds of the human race lasting the next year become 2.5%. Let's spoil ourselves a little and aim for successfully surviving a half a century of such risk; that puts us at the abysmally low survival odds of 2.2e-80%, even ignoring an inevitably increasing population (to be fair, I have been including infants in the 6.7 billion population estimate - the fun of sensationalism!). If we can assume an even stricter 99.99999999999999% cooperation rate per person, then we can almost guarantee a 99% chance of surviving the next half a century (assuming we make it through 2012). We've already made it over half a century since Hiroshima, so perhaps we are getting close with our presumed per-person cooperation rate.

Voodoo math aside, it just seems amazing to me that no properly motivated terrorist or corrupt/bored government has decided to end it all for our species. Perhaps we are a more tranquil species than I thought?