On Elven life-expectancy

I've been thinking about that memorable passage from the Curse of the Noldor in Tolkien's Silmarillion: “For though Eru appointed to you to die not in Eä, and no sickness may assail you, yet slain ye may be, and slain ye shall be: by weapon and by torment and by grief; and your houseless spirits shall come then to Mandos.”

It occurred to me that this fact has rather disastrous implications for Elven immortality. Now of course, in the Silmarillion we see plenty of Elves meet a more or less violent end, getting killed in battle and the like; but what hadn't occurred to me until recently is how sooner or later they would all be likely to die even in a perfectly peaceful and well-ordered society, by sheer accident if for no other reason. If you live forever, sooner or later you will step on a banana peel and break your neck, or walk into an open manhole while texting, or be trampled to death by a marauding circus elephant, all of which means that actual immortality is vanishingly unlikely.

So I figured I'd try to estimate the actual life span of the Elves. The Wikipedia has a useful List of causes of death; after excluding various age- and disease-related causes, which the Elves would presumably not be affected by, I ended up with the following:

Group	Cause	Deaths per 100,000
E.1	Road traffic accidents	19.1
G.1	Suicide	14.0
G.2	Violence	9.0
E.2	Falls	6.3
E.3	Drowning	6.1
E.4	Poisoning	5.6
E.5	Fires	5.0
G.3	War	2.8

That's a total of 67.9 deaths per 100,000 people. Thus your chance of dying from one of these causes in any given year is p = 67.9 / 100,000 = 1 / 1472.75, which suggests that your mean life expectancy is 1 / p = about 1472 years, and the median life expectancy is −log₂ (1 − p) = about 1020 years (see the geometric distribution page in the Wikipedia).

The probability of not dying in a given year is 1 − p, so the probability of staying alive for at least n years is (1 − p)ⁿ. For someone like Galadriel, who was born before the sun appeared and was still alive during the events of the Lord of the Rings, her age in years must be well over 7000 (see this page). For the p we saw above, the probability of living at least 7000 years is only 0.0086, or about 1 in 116.

For comparison: judging by the 2010 data for the worldwide population by age on census.gov, the world population at the time was 6.8 billion; 1/116-th of that is 59 million; and in that year, there were about 64 million people aged 80–84 and 42 million aged 85 or more. So globally, 7000-year-old Elves (or older) are about as common as people in their mid-80s or older.

Suppose you started with a population of 7 billion and they kept dying at these rates. As we saw above, every 1020 years reduces this original population by half; every 7000 years reduces it by a factor of 116. After 7000 years, you'd have about 60 million members of that original population left alive; after 14000 years, you'd have 500 thousand; after 21000 years, less than 5 thousand. At 30000 years, the expected number of living members of that original population is slightly below 10, and by 34000 years it drops below 1, meaning that you should consider yourself lucky if even one of those original 7 billion elves is still alive.

Sure, 30000 years isn't bad — but it's a far cry from true immortality.

Tolkien would have us believe that Elves sooner or later either leave Middle-Earth by sailing from the Gray Havens, or they just sort of fade away. Well, I suppose that sounds more poetic than the sordid truth: road traffic accidents, suicide — and falls! Those damn banana peels! I can almost imagine Morgoth floating through the Timeless Void, munching on a banana with a satisfied chuckle.