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You are in a world where exactly 90% of all people live for exactly 3 years, and exactly 10% of all people live for exactly 100 years.

Aside from what I mention here there is no information that can indicate whether you are someone who will have lived for 3 years by the end of your life or whether you are someone who will have lived for 100 years.

No matter how long you've been alive, or whether you will die after having lived for exactly 3 years or exactly 100 years, your memory lasts for exactly 1 hour, and anything that happened more than 1 hour ago you have absolutely no memory of.

Whether you are someone with a lifetime of exactly 3 years or someone with a lifetime of exactly 100 years a second is the smallest unit of time you can experience and a second feels like the same length of time either way.

From when you pop into existence until you die you look and feel exactly the same, and there is no way to know your age. You also look the same whether you are someone who will die after having lived for exactly 3 years or someone who will die after having lived for exactly 100 years.

You are also conscious for every second of your life and every second of your life feels the same. Basically nothing other than the statistics I gave earlier can give you a clue as to whether you are someone who will have lived for exactly 3 years or exactly 100 years once you die.

From your own point of view what is the probability that once you die you will have lived for exactly 100 years?

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    $\begingroup$ Is that 90% of people ever born, or is that 90% of people currently alive? $\endgroup$
    – bobble
    Nov 12 at 4:28
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    $\begingroup$ Is this the Sleeping Beauty problem? $\endgroup$
    – tehtmi
    Nov 12 at 5:08
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    $\begingroup$ The first chance you get, plant a tree and write your name on the pot.... $\endgroup$
    – Greg Martin
    Nov 12 at 21:34
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    $\begingroup$ @Bass Still irrelevant. The odds of flipping a coin don't change just because you flip many more coins, and the odds of being a 3/100 year old don't change just because people are born more frequently (or less frequently). For all the question cares, every single being could be isolated in a room, unaware of how many other rooms there are. I don't see how any of what you're bringing up actually matters for the question at hand. Birth rate and age pyramids are completely irrelevant. $\endgroup$
    – Flater
    Nov 13 at 8:43
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    $\begingroup$ "From your own point of view" at which point in time? Even if I have no memory of 1 hour ago, I can rescue some memory from 59 minutes or 1 second ago. So, basically I can count all my life long. And every second "feels the same". If I count to ~90 million, I lived 3 years. If I count significantly more, I will (have) live(d) 100 years. $\endgroup$
    – Thomas Weller
    Nov 13 at 11:18

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The frequentist answer to this question is

Either 0 or 1, but you don't know which

This is because

You haven't specified the way that the individual is sampled, so there's no randomness in the problem as it is currently stated. This is not just pedantry - the sampling mechanism is vital to the problem's resolution and different sampling procedures will lead to completely different solutions.

Two careful descriptions of sampling procedures would be:

(1) You start as a disconnected consciousness that is placed in a baby being born that is selected uniformly at random from all babies born that day. You are asked the question after your death. In this case, there is no lifetime bias, and the probability that you lived for 100 years 10%.

(2) You are enrolled in the study on a particular day. Once you die, the investigator asks you the question. In this case, the fact that you were enrolled in the study on a particular day is informative, since short-lived people are less likely to be around to be enrolled. Therefore, in this case, the lifetime bias is relevant and the probability is 79% as discussed in previous answers.

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  • $\begingroup$ For (2), it would be based on the proportion of people alive on that day which (as noted in comments) depends on the historical birthrate. Sneftel's sampling assumption is slightly different; I think it doesn't depend on birthrate. (Florian F calculates based on assuming a constant birthrate which gives the same number as Sneftel, but doesn't seem to acknowledge the distinction.) $\endgroup$
    – tehtmi
    Nov 13 at 4:59
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    $\begingroup$ The question is rooted on the perception of the individual being asked the question, not the individual asking the question. The former's perception does not care about the birth rate. Your answer here is answering the likely distribution among a collection of test subjects, which hinges on the selection procedure of the test subjects, which has no bearing on this one person who has no memory, no knowledge of anything happening outside of the room. You're answering a different question than what's being asked. $\endgroup$
    – Flater
    Nov 13 at 9:06
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    $\begingroup$ @Flater the data point "I've just found out I'm one of the select few being asked a question about my longevity" could play into how one evaluates ones own expected longevity, especially if one knows the sampling methodology. $\endgroup$
    – Martin Kealey
    Nov 15 at 9:40
  • $\begingroup$ @MartinKealey The question bends itself backwards establishing that the person has no meaningful concept of a memory and is therefore answering the question solely based on the provided information. Additionally, as is always the case with puzzles, the person is a perfect logician, i.e. the puzzle's answer is about a correct answer rather than a psychological "what would real people say?" exercise. Within those constraints, focusing on the world outside of the painted scenario and how its scientists sample case studies is just a masturbatory way to avoid the actual puzzle being presented. $\endgroup$
    – Flater
    Nov 15 at 21:43
  • $\begingroup$ @Flater This is not answering the question. It is rejecting the question. Even from a Bayesian perspective, surely we can't purely objectively assign a probability to every possible belief. Is there some definitive answer that you think avoids making this kind of assumption? For example, the next-top-voted answer by Sneftel explicitly makes the strong self-sampling assumption which seems to be the same kind of assumption as what's suggested here; I don't see any substantive difference between presenting such an assumption abstractly vs imagining a scientist. $\endgroup$
    – tehtmi
    Nov 16 at 2:28
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In a sample of 10 random newborns, 1 will be long-lived, accumulating 100 person-years, and 9 will be short-lived, accumulating 27 person-years. So 100/127 person-years are lived by long-lived people. So under the strong self-sampling assumption, the probability is 100/127, or ~79%.

That assumption is

philosophically controversial, but I think you probably agree with it.

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    $\begingroup$ I do agree with the assumption. Maybe it is more convincing if you consider what happens when the short-lived age is just one hour, or no time at all. $\endgroup$
    – Florian F
    Nov 12 at 20:09
  • $\begingroup$ @FlorianF That strikes me as begging the question. $\endgroup$
    – Sneftel
    Nov 12 at 20:54
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    $\begingroup$ Is that first sentence accurate? Grabbing 10 random people isn't going to have a 1:9 ratio, but the ratio of your answer. $\endgroup$
    – rtaft
    Nov 13 at 20:40
  • $\begingroup$ @rtaft Agree, should be "10 random newborns". $\endgroup$
    – GB supports the mod strike
    Nov 14 at 8:17
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    $\begingroup$ @GBsupportsthemodstrike “newborns”, in contrast, is a very well-chosen word and I have stolen it. $\endgroup$
    – Sneftel
    Nov 14 at 8:35
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Let's call B is the yearly birth rate.
The total population is on average B*90%*3 + B*10%*100 people.
The short lived population is on average B*90%*3.
So if you pick any person alive, the probability for it to be short-lived is
27 / 127 = ~21%.

But

But on the other side, as a citizen of the world, you are born with 90% chance to be short-lived.

So...

Which one is correct?

I think that the fact that you are alive is a measurement of your life span. Alive means probably long-lived, as a long-lived person has more opportunities to ask the question of its life-span.

So you have an a priori probability of 90% to be short-lived. But the fact that you are alive and you have no idea how old you are tilts the balance towards being long-lived. And the calculations gives only 21% for being short-lived.

So it gives a 79% chances to live 100 years. By which I mean: if at any time you come to wonder what is your life span, then there is a 79% chance the answer is 100 years.

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  • $\begingroup$ I think you've got a math typo there. $\endgroup$
    – Sneftel
    Nov 12 at 12:06
  • $\begingroup$ I would say I mixed it up badly. I fixed it. Hopefully. $\endgroup$
    – Florian F
    Nov 12 at 14:09
  • $\begingroup$ The birth rate is irrelevant for the question. Your first line results in B times 127. Your second line results in B times 27. When you divide them, the B's cancel out. It's perplexing to me how so many people are assuming that the birth rate is a necessary element here. $\endgroup$
    – Flater
    Nov 13 at 8:53
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    $\begingroup$ Wording matters. It does not "give a 79% chance to live 100 years". From the perspective of the person whose life it is, they were only given a 10% chance to live to 100. How often you wake them up and ask them has no bearing on that. However, if a scientist were to visit every person every hour of their (the person's) life, then 79% of visits would be with a person who will grow to be 100 years old. In other words, if we observed one of these visits at random, there's a 79% chance that this observed person will become 100 years old. $\endgroup$
    – Flater
    Nov 13 at 8:58
  • $\begingroup$ I added an explanatory note to my last statement. $\endgroup$
    – Florian F
    Nov 13 at 10:55
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Assumptions:

This world has existed functionally forever with the exact same ratio. This means that whenever someone dies, they're instantly replaced by another person with an identical lifespan. The population count can only be changed in perfect chunks of 10, which doesn't affect the answer.

I will consider this question as soon as I've forgotten I considered this question (and won't be doing shenanigans to try and remember anything past an hour).

New assumption based on comment below

"90% of all people" means "90% of all people living at any given moment", not "90% of all people being born"

Answer:

I should consider the probability 10% of having lived 100 years by the time I die. I will consider this question once per hour, and so will everyone else, meaning 10% of the observers are people who will die at the age of 100, and the only thing I know is that I'm an observer.

On a side-note:

If we instead assume each person is asked how long they think they lived when they die, I should consider the likelihood 1/301 that I lived 100 years. If we consider a population of 10 people over 300 years, It will have had 9*100 3-year lifespans, and 1*3 100-year lifespans, and each lifespan only contributes 1 observer.

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    $\begingroup$ In the second option, over 300 years you will have 900 short-lived births and only 3 long-lived births. You certainly don't have 10% of long-lived births. $\endgroup$
    – Florian F
    Nov 13 at 16:38
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    $\begingroup$ @FlorianF Hmm, yes, I might have misinterpreted the statement "You are in a world where exactly 90% of all people live for exactly 3 years, and exactly 10% of all people live for exactly 100 years", but even then, I don't think it's unambiguous... $\endgroup$
    – Bearmarshal
    Nov 13 at 17:07
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I will attempt a conversion to the sleeping beauty paradox. Current post -> sleeping beauty:

long life of 100 years -> long sleep of two days
short life of 3 years -> short sleep of one day
90% births (experiments) short life -> 50% tosses (experiments) short sleep
10% births (experiments) long life -> 50% tosses (experiments) short sleep

Now let's go in reverse to see the two sides of the story:

"Halfers" focus on the original experiment probability. The same logic would lead to an answer of "10% to be a longlifer".

"Thirders" focus on the sum of possible events. The same logic would lead to an answer of (100/(100+3)) = "around 97% chance that my current life year is part of a long life".

I'm already treading on the verge of understanding with that paradox, so criticisms are very welcome.

Correction (After peeping Sneftel's answer):

The thirder events should be weighted by probability... So each of the 100 gets a 0.1, and each of the 3 gets a 0.9, leading to (10/12.7) = 78.7%

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90% of people live for 3 years, and 10% live for 100 years.

So, at any given moment, if a 3yr old 'dies' they will be replaced with a new 'to be 3yr old' because the stat of 90% must be constant. Since any given person has the 10% chance of being one of the 100yr olds, and that value is constant, then it will always be a constant 10%.

The Probability is 10%.

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  • $\begingroup$ I don't see how this is different from Bearmarshal's answer $\endgroup$
    – msh210
    Nov 15 at 6:27
  • $\begingroup$ @msh210, I understood he changed his, based on his assumptions being changed. Now, I can see he has the same 10% result, but for different reasons? $\endgroup$
    – Guesser
    Nov 20 at 1:04

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