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Let's imagine there are some ageless people living in big European or American cities. Ageless, forever young but not immortal. They can still die from car crash, gun or knife wound etc.

Assuming crime and accident rates will be the same in future, is it possible to calculate an average life expectancy of an ageless person in a major city of today?

P.S. I understand that there are different cities, let's deal with some average US crime and accident rates to make the question less broad. Let's also assume this person will live normal life — like go to the street, shops, cafes, use transportation, meet people etc.

I found this info but I'm not sure how to calculate the average life expectancy of an ageless guy using it.

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    $\begingroup$ Fun mathematical question! $\endgroup$ – Joe Bloggs Aug 23 '18 at 6:57
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    $\begingroup$ In a fictional setting? No, because you are the author and you decide how the dice roll for your protagonists. If you are looking for averages, such an exercise is pointless because no person ever lives an average life, but instead end up somewhere in a probability distribution. And — again — you as the author decide where in this distribution they end up, especially so when we have just a few individuals. Also: if your ageless protagonists are the least bit clever, they will learn how to defend themselves and avoid danger. $\endgroup$ – MichaelK Aug 23 '18 at 6:57
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    $\begingroup$ But if you insist: here you go. In 2000, the death rate attributed to "other" — under which crimes and accidents are included — is 181 deaths per 100 000 citizens. That means a 0.2% probability of dying each year. This means an average life expectancy of 500 years. In that enormous time, the "other" post in the statistics will change. Because we have no idea at all how crime and accidents will affect us 500 years in the future. $\endgroup$ – MichaelK Aug 23 '18 at 7:10
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    $\begingroup$ Averages are tricky beasts. If Bill Gates would walk in my office now, on average we will all be billionaire. $\endgroup$ – L.Dutch - Reinstate Monica Aug 23 '18 at 7:14
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    $\begingroup$ If you can find age-specific data, it's worth considering how your immortals live. If I'm an ageless 80-something, but I live my life like a 20-something (or a 40-something), I will have a very different accident/crime injury profile than a typical 80-year-old. $\endgroup$ – Cadence Aug 23 '18 at 7:28
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From the statistics page you cited: The odds of dying from an injury in 2014 were 1 in 1,576 according to the latest data available. Let us suppose this is the odds of dying in a given year due to misadventure or violence. This makes it a probability problem.

Figuring out the actual answer to your question - what is the expected value of the number of years you live if this is your only chance of death (i.e. the life expectancy) - is beyond my skill. But I can say this: Your odds of surviving N years in a row is (1 - 1/1576) ^ Nth power.

Your odds of surviving any given year are 1 - 1/1576, and by the Rule of Multiplicaton, the fact that you have to be lucky N times in a row means the total probability is that number to the Nth power.

You can graph this equation here.

This person has about a 50/50 chance of living 1100 years, and a 1 in 10 chance of living about 3600 years or so. It asymptotically approaches 0.

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    $\begingroup$ This is a clever approach. I believe this question is unanswerable, just so that the OP is aware, I'm gonna point out flaws. Please don't see that as an attack, your answer is fine as is. 1) This only takes into account death by injury which is not that likely as anyone who has ever considered casualty insurance might be aware. 2) It assumes that the rate stays constant. I don't believe it was close to constant for the last 1100 years. 3) Age is a factor in risk of dying because of injuries in many ways 4) It greatly depends on your profession. Construction worker or desk job? $\endgroup$ – Raditz_35 Aug 23 '18 at 9:17
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    $\begingroup$ Re: #2 - yes, my answer involves this assumption because the question specifies to assume that it's true. And yes, #4 is quite correct. It greatly depends on profession, and how often you are on the roads, and so on. In general, given the assumption of a fixed rate, this still produces an equation of the form (1 - 1/blah)^N, only the specific value of 'blah' used will change. $\endgroup$ – Dayton Williams Aug 23 '18 at 9:26
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    $\begingroup$ This is a geometric distribution, with parameter p = 1/1576, which is the chance of dying in any given year. The mean of the geometric distribution (average age) is 1/p = 1576 years. The median age is -1/log_2(1-p) = 1092 years; your estimate of 1100 was pretty good. The cumulative distribution function is 1 - (1-p)^k, where k is number of years. If you set that equal to 0.9 (90% chance of dying = 10% chance of living), solve for k=3627, also pretty close to your estimate. Feel free to exit this in /mathwinkemoji (which should exist). $\endgroup$ – kingledion Aug 23 '18 at 11:04
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    $\begingroup$ This assumes that the ageless person is at all times equivalent to a normal person. But their lifetimes' worth of experience is likely to lead them to be more able to prevent coming to harm. They will have considerably more knowledge on seeing other people get hurt, which gives them a better idea on how to avoid common injuries and causes of death. $\endgroup$ – Flater Aug 23 '18 at 14:56
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    $\begingroup$ Well obviously clinically immortal spherical cows won't survive for more than a few minutes, as they're all stuck in frictionless vacuums. (Relevant XKCD: xkcd.com/669 ) $\endgroup$ – Ghedipunk Aug 23 '18 at 22:24
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Let me focus on math part only and explain how your expected life rate depends on the probability of a death in a given year. I will use your assumption that this probability of death does not change so let's assume it is $p$. $p$ has to be greater than $0$ and less than $1$ (that's a general requirement for probabilistic). We'll get back to this value later.

So $p$ is your probability of dying during a specific year if you were alive in the beginning of it and similarly the remaining $1 - p$ is the probability you survive another year.

Now to die at the age of $n$ you need to live for $n - 1$ years and then die in the last, $n$-th year. which means a probability of such event is

$$ P_n = ( 1 - p )^{n-1} * p $$

An expected lifespan is an expected value of a random variable of your age with a probability to reach a value $n$ as $P_n$ calculated above. In other words it is

$$ E=\sum_{n=1}^\inf (n*P_n) $$

Lets put the $P_n$ values into the equation

$$ E = \sum_{n=1}^\inf ( n * p * (1-p) ^ {n-1} ) $$

Let's extract constant so that the power is re-indexed to n

$$ E = \frac{p}{1-p} * \sum_{n=1}^\inf (n * (1-p)^n) $$

Since for $n=0$ the value of $n*(1-p)^n$ is also $0$ (since we multiply everything by $n$ which is $0$ so we can reindex the whole sum to start from $0$

$$ E = \frac{p}{1-p} * \sum_{n=0}^\inf (n * (1-p)^n) $$

Now let's apply a formula to calculate this infinite sum and calculate it and as a result we get

$$ E = \frac{p}{1-p} * \frac{1-p}{(1 - (1-p))^2} = \frac{p}{p^2} = \frac{1}{p} $$

In other words your expected lifespan is exactly $1/p$ where $p$ is probability of death during one year.

Now let's apply various values provided in other answers and comments:

  1. MichaelK's comment gives $p=0.00181$ which accounts to the average lifespan of $552$ years (based on statistics from 2000)
  2. Dayton Williams in his answer gives $p=\frac{1}{1576}$ which accounts to the average lifespan of $1576$ years
  3. Chris Becke's answer gives $p$ between $0.0003$ and $0.0006$ which accounts to the average lifespan between $1667$ and $3333$ years (various statistics, 2010 - 2012)

etc.

Now depending on details you can modify what you include or exclude into your death cause that can kill an ageless person and adapt values accordingly.

This is a pure mathematical approach ignoring few things though:

  • The exposure to dangers of external world are reduced until you are close to being mature. You might add 15 years (or something like that) to your results to cover this aspect
  • The probability of death change over time (ignored as explicitly requested by OP, but in general it decreases over time except a situations like war so it might increase the average lifespan)
  • The ageless person gathers experience that should help them survive even longer, e.g. notice the warning signs and avoid most dangerous situations resulting in a lower death probability than a normal mortal
  • Ageless person may attract various people due to their lengthy life. It might increase their death probability and as a result reduce the lifespan
  • There might be other aspects impacting the lifespan of an ageless person that are not influencing normal mortals and are hard to predict (you have an area to add whatever you think relevant changing this average according to your needs)
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  • $\begingroup$ This is the answer I would have written if I remembered enough of my infinite-series math to do the math myself. If I could transfer the votes from my answer to this one, I would :) $\endgroup$ – Dayton Williams Aug 23 '18 at 17:15
  • $\begingroup$ Well, I wanted to give a fish rather than a fishing rod ;-) Now OP can decide what to factor into "cause of death" that applies to the ageless people and easily coun't himself. Also replacing the $P_n$ with some other infinite series/function representing how the probability of death changes over time one can take the second formula and calculate the infinite sum from that point getting a new result, however there is a risk of math getting trickier. $\endgroup$ – Ister Aug 24 '18 at 7:57
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If the probability of a death happening in one year is 1 in 1576, then the life expectancy is 1576 years.

This all follows the Poisson distribution: the probability of observing $k$ deaths in a year is $p(k) = e^{-\lambda} \frac{\lambda^k}{k!}$ where $\lambda$ is the avarage number of deaths during this year ( 1 in 1576, per person). The expectation value for such a distribution is $\frac{1}{\lambda}$, i.e. 1576 years.

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    $\begingroup$ You've beaten me by 12 seconds, but my answer is a bit more elaborate ;-) $\endgroup$ – Ister Aug 23 '18 at 12:22
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    $\begingroup$ Hehe, hopefully your answer wont assume that everyone feels comfortable with reading mathematical notation $\endgroup$ – Syntaxén Aug 23 '18 at 12:23
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    $\begingroup$ It is but limited to sigma as an infinite sum sign only ;-) I'm also explaining the results so one might completely ignore the "why" (calculation) and just jump to the conclusion. $\endgroup$ – Ister Aug 23 '18 at 12:29
  • $\begingroup$ Of course,, 1 in 1576 is way too low. Still, +1. $\endgroup$ – RonJohn Aug 23 '18 at 12:44
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    $\begingroup$ Well, you could put it like "If the probability of a death happening in one year is 1 in x, then the life expectancy is x" and leave it to the OP to find trustworthy statistics $\endgroup$ – Syntaxén Aug 23 '18 at 12:50
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You don't want to calculate the average life expectancy. It's really meaningless because many individuals are going to live shorter, or much much longer lives. Both depending on luck, and whatever activities they engage in.

The Micromort is a unit that describes 'risk' as a 1 in a million chance of being killed by something.

As you can see from the tables - if you eliminate deaths from natural causes, people are exposed to, between, 300 and 600 micromorts related to unnatural causes per year. This literally means you can, with a population of 1 million immortals - expect to lose 300 to 600 of them to death per year to accident, murder or suicide.

That said - applying the micromort cost of using a motorcycle per km has some sobering outcomes, but even the best forms of mass transit lead to a decimation of your otherwise ageless population over time.

Addendum:

@lster's answer on infinite series is pretty good at transforming 1 in a million chances into your requested "average lifespan" - I think it would be more valuable to consider either the half-life of the population of immortals - the duration over which you expect to loose half of them to death :- which then makes it very easy to ignore the hard math and compute how long it would take to reduce a given starting population to some small number - i.e. how many times do you need to halve the population to get to about the target size.

Other things to consider are immortals would probably value their lives more and consequently engage in less risky activities. And, over a given population there would be a natural filtering effect where immortals (as a group) with risky behaviour would have a shorter half-life and thus be pruned faster, leaving you with a longer lived population of relatively risk averse individuals.

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  • $\begingroup$ How is the average life expectancy meaningless? You have given enough information to construct a poisson distribution. Which can offer all the information OP asked for. 5%ile, 25%ile, 50%ile (average), 75%ile, and 95%ile life expectancy. $\endgroup$ – user53870 Aug 24 '18 at 13:23
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I'd say it would be much much longer than anything calculated here, since when you can theoretically live forever, people will be much more careful, homicide would be punished much much harder and generally, safety would be a much bigger concern.

So just make something up.

[As an interesting sidenote: Drugs with direct physical harm lice Alcohol, Tobacco, MDMA etc. would be far less popular, since they would be the only way to actually "age" by destroying your body unnaturally but slowly]

[Edit due to the comments: It does make sense that some/most people would value life even less, resulting in what @Clay Deitas said. However there would also be SOME people who realize the potential of immortality and would take extreme care of themselves, leading to a two-class society separated by nothing but mindset.

"The Immortals" as I call them would naturally have a high interest in the "YOLO" Group being as large and as risky as possible to increase their status as immortals and also keep the population down. They could for example introduce measures to make people age again (by a virus or so).

They would certainly be at the top o society, an have no problem with plans that take a century to set up, leading to a very interesting social dynamic.

EG: you can have your "average lifespan" be directly controlled to fit your plot by a council of lets say the 50 most long-lived people]

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  • $\begingroup$ If everyone on earth was ageless we would have looser laws and take bigger risks because we would not find life as precious and would need a way to cull the population. We might not even have hospitals anymore. $\endgroup$ – Clay Deitas Aug 23 '18 at 11:32
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    $\begingroup$ If people really worked that way, then 20yr olds would be much more careful than 60yr old people, since they've got much more to loose. In fact, it's the other way around: teenagers think they'll never die, which makes them more careless. $\endgroup$ – nikie Aug 23 '18 at 11:35
  • $\begingroup$ Upvoted because I think there would be an effect one way or the other. I don't think we can just extrapolate from current violent-death statistics in our society. Road-safety regulations might be tighter, and/or some people might be less willing to travel by car. $\endgroup$ – Peter Cordes Aug 23 '18 at 11:51
  • $\begingroup$ @ClayDeitas Why would there be a need to cull the population? If you can only die from injuries, then less food/water/living space will hardly affect you. As Thanos from the Avengers' Infinity War points out, a large population usually dies due to too much competition for limited resources such as food and water. But, if you're ageless, then these don't pose any concerns, and so a large population would hardly affect you. $\endgroup$ – Adi219 Aug 23 '18 at 12:36
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    $\begingroup$ Hey man, you wanna buy some lice? $\endgroup$ – Mazura Aug 23 '18 at 20:22
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I suggest looking at this from a different angle. For ageless semi-mortals, what are your risk factors? Also, what is the risk tolerance of someone who could potentially live forever? I suggest that both of these are tied directly into an individuals level of happiness and social connectivity, and your life expectancy will be very strongly correlated. Let me explain...

Facing eternity when you are discontent, lonely, or depressed means you are much more likely to take on risks that could improve your happiness. Maybe that means you take up skydiving as a hobby or start working a more dangerous job to increase your income (in this kind of society, I can almost guarantee that physically risky manual labor jobs will pay relatively much higher wages).

If you are in a loving relationship, have lots of friends, and a happy disposition, eternal happiness seems a very serious thing to risk by doing any kind of risky behaviors. That means you are more likely to live below your means until you find a suitably safe job. Also you will more likely forgo the sky diving and maybe take up board games. You may not have as much fun (perhaps), but you will live longer and dying would leave behind people who care about you to live for a long, long time.

Actuaries constructing their death tables will do so based primarily on these factors. Someone living blissfully should expect to live indefinitely. Due to survivor bias, society would slowly become happier and happier, until all the unhappy people die and those that remain are likely to live for eons, dying only due to disaster.

(As a side note, any numbers connected to death by misadventure in today's society are almost certainly lower than they would be if you simply removed disease from the equation, all else being equal.)

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This is interesting mathematical question.

If we are able to gather the data of deaths and its causes and classify them as "Natural Death" or "External forces". Using this dataset , we would be able to able to calculate the "probability" of human to die at a particular age due to external forces.

How long a person will live ? - Probability may tell us the chances of humans to survive at a given time, this will be same for all humans in a data set. If we attribute the external causes to death, then different age group will have different probability, but here people are ageless, so everyone has equal chances.

Although, average age of a population can be found over a period using "historical data/current". It is illogical to find average over a probable data set.

Example: 100 out of 20 die of natural death. So, a person has probability if 0.8 to survive at any moment/age. Average age - we could , find average age if we are given current/historical data-set, not future.

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  • $\begingroup$ your distinction between natural death and external forces seems arbitrary. Can you be more precise? $\endgroup$ – L.Dutch - Reinstate Monica Aug 23 '18 at 8:09
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    $\begingroup$ Hello and welcome to Worldbuilding. This post does not answer the question and reads more like commentary on the question. I suggest that you complete the reasoning you have started in the post and bring it to its conclusion, making the calculations that you have described. Until that is fixed, I am voting -1. $\endgroup$ – MichaelK Aug 23 '18 at 8:26
  • $\begingroup$ What I mean by natural death here is, deaths caused by old age, body cells/organs start malfunctioning, organs getting weaker. As body gets old - there are numerous changes that undergo in our body. Of course our body's functioning is affected by other external factors as well in part. - Food intake, climate. But, there are certain changes which happen in body as we age, regardless external factors. $\endgroup$ – Ratnakar Chinchkar Aug 23 '18 at 9:13
  • $\begingroup$ External factors as mentioned in question description - includes crimes, climate changes, natural disasters etc. Problems which won't occur in an utopian society/nature. $\endgroup$ – Ratnakar Chinchkar Aug 23 '18 at 9:15
  • $\begingroup$ you see, most of the countries have published avg population of their populous, and predicted the average age for next years too. The data set includes previous death rate, birth rate, death rate by age group. Example, 1. 20-30 age group falls more in death caused by crimes. 2. Age group 65+ is more sensitive to climate/environment changes. These factors narrow down the data set, and can make somewhat probable average predictions. But, if don't age , ageless, every human will be equally likely to be dead by these factors. This'll lead to making our predictions very broad and highly improbable. $\endgroup$ – Ratnakar Chinchkar Aug 23 '18 at 9:30
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Further information:

https://www.statista.com/statistics/241572/death-rate-by-age-and-sex-in-the-us/

The smallest death rate is for females 5-9, who have a death rate of slightly more than 1 in 10,000. Females 25-29 have a death rate about six times as high, and males 25-29 death rate about fifteen times as high, or more than twice females of that age range. It's reasonable to assume that the increase in death rate from 5-9 to 25-29 is not due to people dying from old age, and the difference between men and women is more likely behavioral rather than directly due to biology (although there may be biological influences on behavior). Which of these numbers is more applicable to your scenario is a matter of opinion. Currently, the biological, psychological, neurological, and social aspects of adolescence lead to behavior that increase the death rate, and once people are past that stage, they then start to death with aging issues. Perhaps in your scenario, a large portion of people survive past adolescence and return to 5-9 levels of behavior-induced deaths, while avoiding age-induced deaths.

Thus, I would say the optimistic expectation is that the culture would develop more cautious mores, and have a death rate around 1 in 10,000, while a less optimistic expectation would be 1 in 1000 (the average death rate for young adults).

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There's a number of answers here that show good math, but there's also a social aspect to the case.

Young males are most likely to die violent deaths, because they're most likely to be violent criminals. Even in an ageless society, this will probably remain this way, and pull down the average age a lot much like infant deaths did the european average for a long time (basically until the medical science got up to speed). In the middle ages, if you made 20 you'd probably make 50 or even 60 even if the average lifespan was around 40, simply because if 33% of all children die, that pulls down the average.

Following this, there'd probably be a death spike in the age bracket under 40, and those who survive that are pretty good at surviving, and will survive really, really long.

That said, you also need to change how minds work a bit. As we get older, it's harder to learn to adept to new technologies - take for example the average grandmom's grasp of computers. The reason the amount of computer savvy people increasing as much as it does is because those who don't are typically older and first do die. So you'll also need to enable your ageless people to learn a life long, if you want them to be any kind of relevant after, say, 100 years.

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