Am I Going To Die This Year? A Mathematical Puzzle

A few years ago, physicist Brian Skinner asked himself: What are the odds I will die in the next year? He was 25. What got him wondering about this, I have no idea, but, hey, it's something everybody asks. When I can't wedge my dental floss between my two front teeth, I ask it, too. So Brian looked up the answer — there are tables for this kind of thingand what he discovered is interesting. Very interesting. Even mysterious.

Obviously, when you're young (and past the extra-risky years of early childhood), the chances of dying in the coming year are minuscule — roughly 1 in 3,000 for 25-year-olds. (This is a group average, of course.)

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But eight years later, the tables said, the odds will roughly double. As Brian writes in his blog post, "When I'm 33 [the chances of my dying that year] will be about 1 in 1,500."

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And eight years after that, he says, the odds double again: "It will be about 1 in 750."

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And eight years later, there's another doubling. Looking down the chart, you'll see that keeps happening and happening and happening. "Your probability of dying during a given year," Brian writes, "doubles every eight years." Hmmm. When I looked at the latest tables (Brian's came from 2005), he's more or less right. Which got me wondering ...

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Why eight? Why the doubles?

This wasn't Brian's discovery. A British actuary, Benjamin Gompertz, noticed this pattern back in 1825, and ever since it's been called the Gompertz law of human mortality — yes, death creeps closer, but it creeps closer in orderly steps (for humans about every eight years).

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Doubling of this sort, when plotted on a chart, looks scary in the later years, but every interval early in the curve is also a doubling. So the same thing keeps happening, only the effects become more pronounced. Anyone reaching the age of 100 seems to have a 1 in 2 chance of getting to 101.

Looking at his pattern, Brian writes, "I can say with 99.999999 percent certainty that no human will ever live to the age of 130." (That's assuming, which one shouldn't, that we have no new, heroic medical advances.)

OK, so this happens. The pattern, says Brian, "holds across a large number of countries, time periods and even different species. While actual average lifespan changes quite a bit from country to country and from animal to animal, the same general rule that 'your probability of dying doubles every X years' holds true."

But here's the dangling question: Why the regular interval? Why eight years for humans?

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Brian's answer: "It's an amazing fact, and no one understands why it's true."

Really? Shouldn't there be some obvious explanation?

It's pretty obvious that when surveying a large population, death is not really a random, sudden bolt of lightning out of the blue. If it were, as Brian points out, the bolt would hit randomly, and in any collection of people ...

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... the babies would be as likely to die as the oldsters, youngsters, middle-agers. But that's not how it works. Older people die more frequently than younger people (in peacetime, anyway).

So — random, death isn't.

Couldn't the latest biological explanations for aging explain an eight-year doubling pattern? Brian considers this question in his essay. He calls it the "cops and criminals theory." (It's based on a short paper by Boris Shklovskii.) As Brian describes it:

Imagine that within your body is an ongoing battle between cops and criminals. And, in general, the cops are winning. They patrol randomly through your body, and when they happen to come across a criminal, he is promptly removed. The cops can always defeat a criminal they come across, unless the criminal has been allowed to sit in the same spot for a long time. A criminal that remains in one place for long enough (say, one day) can build a 'fortress' which is too strong to be assailed by the police. If this happens, you die.

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Lucky for you, the cops are plentiful, and on average they pass by every spot 14 times a day. ... But what happens if your internal police force starts to dwindle? Suppose that as you age the police force suffers a slight reduction, so that they can only cover every spot 12 times a day? ... The difference between 14 and 12 doesn't seem like a big deal, but the result was that your chance of dying during a given day jumped by more than seven times. And if the strength of your police force drops linearly in time, your mortality rate will rise exponentially.

This is the Gompertz law, in cartoon form: Your body is deteriorating over time at a particular rate. When its 'internal policemen' are good enough to patrol every spot that might contain a criminal 14 times a day, then you have the body of a 25-year-old, and a 0.03 percent chance of dying this year. But by the time your police force can only patrol every spot seven times per day, you have the body of a 95-year-old with only a 2 in 3 chance of making it through the year.

This sounds right, that our immune system deteriorates at a steady pace, leaving us with fewer and fewer cops to remove the troublemakers in our bodies. As a metaphor, it works. But, says Brian, "unfortunately, the full complexity of human biology does not lend itself readily to cartoons about cops and criminals." There is no biological finding that explains the eight-year pattern we find in the mortality tables. The idea is nice. But the math? It has no obvious logic, no explanation — not yet. We know death is approaching, but why does it like the number eight?