How long will an ageless person live? [duplicate]

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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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9 Answers
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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) ^ N^th 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 N^th 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.

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

$$
E = fracp1-p * sum_n=1^inf (n * (1-p)^n)
$$

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 $0$

$$
E = fracp1-p * sum_n=0^inf (n * (1-p)^n)
$$

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

$$
E = fracp1-p * frac1-p(1 - (1-p))^2 = fracpp^2 = frac1p
$$

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:

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:

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 $p(k) = e^-lambda fraclambda^kk!$ where $lambda$ is the avarage number of deaths during this year ( 1 in 1576, per person). The expectation value for such a distribution is $frac1lambda$, i.e. 1576 years.

1 in 1576

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.

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]

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.)

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.

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).

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.