For How Long Can a World with Nuclear Weapons Avoid Nuclear Annihilation?

We live in a world with nuclear weapons. We know there are enough nukes to blow up the world many times over. And we know that the more time passes the more opportunities there will have been for nuclear war to break out.

But how much time will have to pass for this possibility of nuclear war to become a near certainty?

Let's make four assumptions:

1. Only 10 governments have nuclear weapons at any given time.
2. A government exists for four years on average.
3. Only .1% (1 out of 1,000) of governments with nuclear weapons are stupid or suicidal/homicidal enough to accidentally or intentionally start a nuclear war.
4. Nuclear war will escalate into nuclear annihilation (either directly through the blasts or as an indirect effect in the form of fallout and nuclear winter).

Please note that while these assumptions seem pretty reasonable they are of course hardly proven facts. For example, maybe a nuclear war between two countries can remain relatively regional and contained instead of escalating into total nuclear annihilation of the world. Or maybe the risk that any given government with nuclear weapons will start a nuclear war is much greater than one in a thousand. Or perhaps over time there will be many more countries and even non-state actors who will acquire nuclear weapons.

So if you think some of the assumptions above are incorrect, just adjust the numbers accordingly. For now, given the four assumptions above, how long would it take for the probability of nuclear annihilation to approach .99?

Probability Calculation
Assumption #2 & 3 mean that the probability that a government with nukes will not start a nuclear war in any given four year period is .999. Assumption #1 says there are 10 such governments at any given time, so we get a .999 ^ 10 = 0.990045 probability of survival per four year period.

How many periods will it take for this probability to go down to .01?

.990045 ^ x = 0.01
x = 460.29

Each period is four years, so we get 4 x 460.29 = 1841.16 years before we or any other civilization with nuclear weapons will have almost certainly been wiped out.

The Future
This suggests there are hard limits to the duration of any civilization in which nuclear weapons exist. Such a civilization can't expect to grow, develop and innovate for tens or hundreds of thousands of years. Nuclear weapons put a hard limit to what any civilization could hope to achieve before it's wiped out.

But 1841 years is a long time, so from our point of view, who cares? Après nous le déluge.

Well, what about the probability of nuclear annihilation in our lifetime? Or our children's?

The probability of nuclear annihilation in any given 50 year period is close to 12%.

For a 100 year period it's about 22%.

That's really high.

The Past
The US first developed nuclear weapons in 1945, so the world has had nuclear weapons for over 74 years now. We've survived so far but how great was the risk?

For a short while the US was the only country with nuclear weapons but soon other countries followed. Currently there are 9 nuclear countries.

Let's simplify and say that on average in the past 74 years there were 5 countries with nuclear weapons at any given time. That means that there would have been close to a 9% chance of nuclear annihilation in that period.

A 9% probability does not even seem that high considering how close the world has already come to nuclear war, either by accident or intentionally:

The 1962 Cuban missile crisis is generally represented as a dangerous standoff resolved by sober diplomacy. In fact, it was a single man — Russian submarine commander Vasili Arkhipov — who countermanded orders to launch a nuclear torpedo at an American destroyer that could have set off a full-scale nuclear exchange between the Soviet Union and the United States.

There were numerous other incidents that brought the world to the brink. On a quiet morning in November 1979, a NORAD computer reported a full-scale Russian sneak attack with land and sea-based missiles, which led to scrambling U.S. bombers and alerting U.S. missile silos to prepare to launch. But it turned out there was no Soviet attack — just an errant test tape.

Lest anyone think the incident was an anomaly, a little more than six months later NORAD computers erroneously announced that Soviet submarines had launched 220 missiles at the United States. This time the cause was a defective chip that cost 49 cents — again resulting in scrambling interceptors and putting the silos on alert.

What Does This Mean?
Nuclear war is barely a topic of conversation in political life. That's crazy. The impact of nuclear war would be truly apocalyptic and the probability that nuclear war will actually occur in the not so distant future is really high. That makes the threat of nuclear annihilation by far the greatest problem faced by mankind.

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Further reading
- Andrew Cockburn's How to Start a Nuclear War: The checks in place to prevent unauthorized people from starting a nuclear war don't work very well.

- William Langewiesche's How to Get a Nuclear Bomb: It is not impossibly difficult for terrorists to acquire nuclear weapons.

- John Hersey's Hiroshima: Classic 1946 New Yorker account of the horrors people in Hiroshima experienced the day the atomic bomb was dropped over their city.

- Daniel Elsberg's Doomsday Machine: First hand account of the dangers and madness of America’s 70 years long nuclear policy.

- Conn Hallinan's There Are Thousands of People Who Could Launch a Nuclear War: Thou-sands.

- Conn Hallinan's A Global Nuclear Winter: Avoiding the Unthinkable in India and Pakistan: Even a "limited", regional nuclear war has catastrophic consequences for the rest of the world. And such a regional nuclear war is much more likely than you think.

- Conn Hallinan's We May Be At a Greater Risk of Nuclear Catastrophe than During the Cold War: "Astounding increases in the danger of nuclear weapons have paralleled provocative foreign policy decisions that needlessly incite tensions between Washington and Moscow."