## Email #26 Explained

A number of people expressed interest in knowing where the numbers in email26 came from, so I've developed this page to explain them.

In that email, I stated that even if someone is optimistic enough to assume that nuclear deterrence could be expected to work for a thousand years before failing, a child born today still would have roughly a 10% chance of not living out his or her natural life. There are several levels of explanation and starting at the highest level first, that's because an 80 year expected life span divided by the 1000 years we were assuming deterrence might work is 80/1000 = 8%. So why did I say 10%? That's because the 1000 years was a ballpark figure and giving the answer as 8%, instead of rounding it to 10%, would imply a greater level of accuracy than is warranted. In engineering, this is called an order of magnitude estimate.

Drilling down to the next level of complexity, why did I divide 80 (or, rather, the rounded number 100) by 1000 to get the risk? If deterrence can be expected to work for 1000 years, there's one chance in 1000 (0.1%) of it failing each year. Or, looking on the brighter side of the coin, there's a 99.9% chance of surviving each year. Then the chance of surviving 100 successive years (again, rounding 80 to 100) is 0.999 raised to the 100th power, which equals 90.48%. But, again, that level of accuracy is unwarranted and 90% is a more meaningful number. If there's a 90% chance of surviving 100 successive years, there's a 10% chance of not surviving 100 successive years.

The fact that the risk (10%) is just the annualized risk (0.1%) multiplied by the number of years (100) is an example of a general rule in probability: If a coin has small probability p of showing Heads (corresponding to a nuclear war occurring in any given year) and we toss it N times (corresponding to the N years of deterrence not failing) then the probability of getting at least one Head in the N tosses is approximately the product Np, provided the product is small.

There are more levels of mathematical complexity we could go into, such as independence and time-invariance, but I'll spare you those discussions. Purists can resolve those two issues by reading my paper (1.8 MB PDF). Instead of more math, let's turn to some issues of a different nature.

The above calculations assumed that everyone died if deterrence failed, even though there are some failure modes, such as a nuclear terrorist incident, a regional nuclear war (think India and Pakistan), or a limited nuclear war betwen the US and Russia, in which fatalities would be much lower. But, for reasons explained in my paper, I was using a complete failure of deterrence. That's OK so long as the 1000 year estimate is also referenced to a complete failure. Since most people think of deterrence in terms of Mutually Assured Destruction or MAD, it is reasonable to use a complete failure (Mutually Assured Destruction) in these calculations.

There are other levels of complexity that we could go into, such as what fraction of the population would die in a complete failure of deterrence, but I'll spare you both the horror and the time. If you have the time and can stand the grim statistics, that too is covered in my paper.

Continuing with the assumption that deterrence could be expected to work for roughly a thousand years before failing, I also stated that

• the risk would be equivalent to having your home surrounded by thousands of nuclear power plants;
• the risk would be equivalent to sky diving twice a week, but with the whole world in the harness;
• nuclear war would be the third most probable cause of death for a child born today.

The first comparison comes from the fact that nuclear power plants are designed with at least a million year Mean Time to Failure (MTTF). Again, that's a catastrophic failure as happened at Chernobyl, not a partial failure as occurred at Three Mile Island. TMI's containment vessel prevented that partial failure from becoming a catastrophic failure. Chernobyl was a much more failure-prone design, without a containment vessel. Living in the middle of a complex with 1,000 nuclear power plants each with a million year MTTF would result, on average, in a catastrophic failure every thousand years – our assumed MTTF for deterrence. Since modern plants have even higher MTTF's, it would take thousands of such plants to equal the "thousand year" assumed risk from nuclear weapons.

The sky diving comparison comes from the fact that one in 100,000 parachute jumps results in a fatality. If nuclear deterrence could be expected to work for 1,000 years before failing, it would take 100 jumps per year (two per week) to equal the risk from nuclear weapons.

Turning to the last bullet, I used the Centers for Disease Control (CDC) list of the leading causes of death in the US. Recalibrating their data (which lists annualized deaths per 100,000 population) in terms of an annualized percent fatality rate we get:

• Diseases of heart 0.2%
• Malignant neoplasms 0.2%
• Cerebrovascular diseases 0.05%
Under the "thousand year" assumed risk from nuclear weapons, a person has one chance in a thousand each year of dying from that cause. That is 0.1% per year and would replace cerebrovascular diseases as the third leading cause of death. Clearly, this would not true every year. Rather, it would be true averaged over time. Most years, no one would by killed by nuclear weapons, but one year there would be a huge spike.

My email also noted that if, instead of a thousand years, we expect nuclear deterrence to work for a hundred years before failing:

• a child born today would have less than a 50% chance of living out his or her natural life;
• the risk would be equivalent to having your home surrounded by tens of thousands of nuclear power plants;
• the risk would be equivalent to sky diving three times every day, but with the whole world in the harness;
• nuclear war would be the most probable cause of death for a child born today, exceeding even the risk heart disease several times over.

All except the first number are obtained just by multiplying the previous risk factors by 10. For example, the two parachute jumps per week becomes twenty, which (rounded) is three per day. The 50% number in the first bullet is a bit more complex because tossing a coin which has a 1% chance of showing heads 100 times (the rounded life expectancy of the child) does not have a 100% chance of showing at least one head. Rather, the answer is

1 - 0.99^100 = 63%

and the chance of the child not being killed by nuclear weapons would be 37%.

Most people I've talked with think that nuclear deterrence is likely to work for no more than 100 years before failing in a final, cataclysmic war. But some people have much higher estimates, with one person saying he thought it would work for at least a million years. Both the Appendix to my paper (1.8 MB PDF) and my article "Soaring, Cryptography and Nuclear Weapons" explain why a thousand years is the longest we might expect deterrence to work, and a hundred or less is probable.

I hope this helps explain where the numbers came from – and why I didn't go into this much detail in the email!

Martin
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Martin Hellman