The reason that I likely drew ball number 7 from the ten ball urn rather than the million ball urn is that the million ball urn had a lot more balls that I could have drawn making it less likely for me to have selected a number less than 11. It works because the other 999,990 balls were available for drawing. If I could reach only balls 1 to 10 in either urn then drawing ball number 7 would tell me nothing about which urn I selected. Since people who haven't been born yet aren't available for drawing, the selection of me here and now tells us nothing about how many more people are likely to be born.

Yes, but to then add the aspect of only looking at pre-selected values of M no longer matches the Carter situation.

The Carter hypothesis does not select a given "M" and so this argument does not apply.
The fact that "the plainly obvious statement that 90% of beings live in the last 90% of any set from which they are generically chosen" is the power of the Carter hypothesis when simply combined with an exponential population curve and assumption that all known species have a finite existence. That is the strength of the argument - that a statistical estimate can be made with so little input. It is also the weakness of the argument because it requires one to ignore any additional knowledge.


Well, it certainly is important to include knowledge into an probability calculation if such knowledge is available to improve the validity and reliability of the calculation. Certainly, one's calculation is only as good as the assumptions that went into the calculation. If knowledge is available but not used, then any such calculation is limited by the assumptions that were used (are the dice loaded?)

If you chose not to use any other information to help describe the likely population distribution (and have a value of M as a result of an essentially random selection {not-pre-selected} from a finite population), then the Carter estimate is the best you can do - but it is fatally weak because you are purposely not using important information that may have significant impact on the likely population curve and expected specie longevity.
(It is like expecting the normal odds of getting a 7 is a valid statistical argument even when you know that the dice are loaded but you just choose to ignore that information and calculate odds assuming that you don't know they are loaded).

I see it as precisely the Carter situation. The "pre-selected value" of M is our own birth number, the scale for the predicted continued longevity of humanity that leads to the "catastrophe". In my game, the selection of M is the same as the selection of "you", out of the N humans that will live. It is identical to the Carter argument, phrased in terms of birth number (which is the only valid way to phrase it without assuming strange things about exponential growth that never hold in any system).
See above.
But that "combination" with an exponential curve is kind of ridiculous anyway. There are hosts of systems that undergo temporary exponential growth, then settle into some other pattern. To use that predictively is to be ignorant of a vast fraction of reality.
The argument falls completely flat even in the absence of any additional knowledge. It makes assumptions that are unjustified, and its own assumptions are its weakness.
Agreed, this is normally part of the value of using probability. But it doesn't "change the probability", it merely changes the inputs we choose to include. The calculation is always purely our own choosing, it has no "reality" to it, and so cannot "change" in some absolute kind of way.
Yes, I agree.

No, you cannot even do that well, it's just wrong probability to apply it to classes of similar M.

It would seem that the argument could never predict an early doomsday because you must use only the birth number as data and ignore any additional knowledge. But the fact that the population is increasing geometrically is additional knowledge and it can't be ignored because the early doomsday prediction depends on it.

I will repeat one last time - that is not what the Carter probability is doing. It does not apply probability to multiple values of a specific "M" to estimate total "N." It applies a statistical argument to any one individual. There is no dependence on the value "M" for the argument.

It simply uses the "the plainly obvious statement that 90% of beings live in the last 90% of any set from which they are generically chosen" and combines with our observed exponential growth to arrive at a simple statistical probability that the end is likely near.

If you use no other information, it is a valid statistical estimate. If you think there are other arguments for probable human life existence, that the shape of our population curve will change in the future, or that your specific birth order is somehow special (not generic), then the simple, Carter (no additional knowledge) estimate will not be accurate to the extent you believe your other knowledge is informative and/or likely to be correct.

Audios.

What you are failing to recognize is that applying it to an individual is invoking an M value-- it is simply the birth number of that individual, and it is used to specify the entire catastrophe scenario. Let me put it another way. Let's say there are a million different intelligent species in the universe, all of whom think of the Carter catastrophe at some point. Let us go into the minds of the subclass of those individuals whose birth number is, say, 15 billion, which is kind of like our own. Now let us ask a simple question: do we expect that 90% of that subclass are living in the last 90% of the number of their kind?

The answer to this, if you understand probability, is: no, we simply have no idea what that fraction will be, which is easily verifiable by simply choosing some distributions of N for those million species and checking my claim that it will not yield 90% (try it, really, you probably can't understand what I'm saying until you do). Saying that 90% is therefore the best we can do is like saying that anything can either happen or not, so everything has a 50% chance of happening if we know nothing else. That's wrong use of probability, which is different from correct use of probability in the absence of much information. The Carter argument, if it is applied to us at this moment in human history, i.e., to this M, is simply wrong use of probability.
I already addressed the fallacy of "combining" that with any expectation of continued exponential growth. On what basis would we choose that combination, pray tell? But that's another issue, I prefer to think of the Carter conjecture in terms purely of birth number, for that requires less absurd assumptions and is therefore wrong for far more subtle reasons.
That's like saying that if I think of a number, and you try to guess it, you can either be right or wrong. Since there are two possibilities, and you have no other information to go on, the best you can do is assume you have a 50% chance of being right. Shall we call that a valid statistical estimate, and blame our lack of information? No, when we lack all information, we also lack the very meaning of probability.
So Carter is using no information to say it will continue to be exponential, but I'm bringing in something new if I ask for a reason to accept that assumption? Not.

This thread appears to be returning to dormancy, so I'll summarize the interesting issues that have emerged. It is a point of some frustration to me that I have made the following mathematically bulletproof arguments yet objections have been raised that don't seem to understand the fundamental points I'm making, so I'll try one last time to condense them into the best and purest form I can, and the reader can take it or leave it as they like. I personally think there is a lot to be learned about what probability can be used for, and even more interestingly, what it can't.

Bottom line: there are two very different ways to state the "Carter catastrophe" conjecture, and they are both examples of bad probability, even in their purest forms with no consideration of whether or not our asking the question itself changes anything (which is a severe problem for the conjecture but I don't need it). Here are the two ways, and what is wrong with them:

1) the conjecture stated in terms of birth number: This goes something like this. 90% of humanity will have a birth number that will ultimately prove to be in the last 90% of humans born. Thus any human may expect with 90% certainty that they fall in the last 90%. That is true as stated, but it is no longer true if that human uses their own birth number (let's say ours is about 15 billion) to infer a coming "catastrophe" in the next 150 billion (to achieve the incorrect 90% likelihood). I have referred to this as "using the M value" to make predictions about N. Others have claimed that is not part of the "catastrophe" scenario, but they have not justified that claim with any suggestion of how there is a "catastrophe" without it. You see, the only way to not invoke M is for a human who has no idea what their birth number is, even vaguely, to then say "I'm 90% likely to be in the last 90% of humans". Note that is perfectly true, but see the important difference? Where's the catastrophe! If the person's birth number might be 100 trillion, for all they know, then humanity could easily live to a quadrillion. Or a million times that-- this is what it means to have no idea what your birth number is, and is clearly violated by the Carter logic. If any use is made of our actual birth number around 15 billion, all bets are off-- the probability argument is simply false at that point, as is easily verified by choosing any arbitrary distribution over N that you like.

2)the conjecture stated in terms of exponential growth rate: Here it goes something like, since humanity is growing exponentially with an e-folding time of (let's say) 50 years, then a version of the birth number argument states we are 90% likely to be within about 2.4 efolds of the end of humanity, say less than 120 years. This version does not require an M value, so takes advantage of the magnitude-free form of an exponential distribution, but it fails for far less subtle reasons. After all, who in their right mind would think it is a valid expectation that if humanity has been e-folding every 50 years for the last handful of centuries, that it should continue to do so for the next handful? In five minutes I could list 100 contradictions to that assumption from everyday experience, it's no better than the pseudoscientific arguments used by creationists (indeed, they do use invalid extrapolations all the time). All you can say is that any exponential distribution cannot be generically expected to extrapolate more than a few efolds into the future, to which I say: duh. But that's not a "catastrophe", merely an expectation that the magnitude-free form of human population growth should be expected to change fairly soon. Again, duh. (Note that the argument goes through perfectly well if the growth slows to, say, an efold of 100 years-- no catastrophe there. We can only be said to be generically chosen from the subclass that shares our growth rate-- unless we make specific reference to M to obtain a catastrophe prediction.)

To me, the Carter Catastrophe reads like wishful Christian thinking wrapped in statistical pseudoscience. It's based on the assumption that we, ie: you or me or anyone alive today, has the equal (or in fact, much greater) probability of living in a time of greater population (hence, a possible distant future). Statistically speaking it is sound...if one completely ignores reality (as, unfortunately, so many of these type of arguments do).

The reality it is ignoring is simply this...I'm alive now (or you are alive now, or my wife is alive now...) because there is no other time that any of us could be alive.

That almost sounds circular or anthropic in nature, but I assure you it's not. Rather, I'll call it the Bio-historical rebuttal.

What I mean is this: who I am is a product of my precise biology--beginning, most fundamentally, with my genetics, which forms the foundation of 'me'. From that, of course, follows the environmental factors that fine-tuned my personality. But the genetics, the precise combination of that one sperm and egg are the ONLY combination that could have produced 'me'. Any other combination, even from my biological parents (as those of you with siblings clearly know) would not and could not produce 'me'. Obvious, my parents are also the results of similar events as were their's, back through time. From this it is trivial to see that, not only could I only be born to those parents through that particular combination of genetic material but, therefore, 'I' could only be born in this particular time.

With this understanding, the foundation of the Carter Catastrophe completely crumbles because there is 0% chance of 'me' (or anyone else alive today) being born at any other time, regardless of how many more generations yet may live.

Therefore, at best, the Carter Catastrophe is an amusing game of statistics, at worst...well, maybe that could be the topic for some interesting novels.

Goodness! Six pages of argument over something so simple to refute! Oy!!
Here, let me end this silliness once and for all.


Let's turn it into a simple logic problem. You have 2 urns: A and B which contain some numerical value within. There are two numerical values of (X) 10 and of (Y) 1 million, which you must connect to A and B in order to find out if A is greater than or less than B. You will do this by pulling a single numerical value (N) out of urn A.
IF N is greater than 10, THEN N falls outside the bounds of X, THEREFORE A must equal Y
ELSE IF N is less than 10, THEN N falls within the bounds of both X and Y, THEREFORE A = ?

if you suggest anything other than "inconclusive" you would be making an illogical statement. Regardless of probability or statistical likelihood, as long as A could be either X or Y, A cannot be either X or Y to the exclusion of the other. Without more information, there is no way around this. Sorry.


Nobody disputes that a small population is a population in danger. This does not mean you can take that population number, by itself, and suddenly start making predictions about extinction with it. The world doesn't work that way.

If I draw a ball numbered less than ten then I don't know which urn it came from, but that doesn't mean equal probability for either. The real question is whether or not drawing balls from urns is the same problem as selecting myself as a representative of all the people who will ever live. I think that it's not.

__________________
Life is like a box of chocolates. All of your choices are bad for you.

I believe this is called thread necromancy.

In any case, sorry, but your logic example is not clear so, at least to me, you haven't ended all this silliness once and for all.

There seems to be room for a lot more silliness to me.

__________________
Life is like a box of chocolates. All of your choices are bad for you.

Let us attempt to address the "silliness" using a very concrete construction. Imagine a supra-intelligent alien species that is the last surviving intelligence anywhere in the observable universe. A statistician from the species has access to all knowledge of every intelligent population that has come before. Specifically, they know the distribution N(n), where N is the number of populations that gave birth to n members before it went extinct.

Now, let us further stipulate that this statistician uses a random process to select one individual from all those populations, where each individual was equally likely to be the one selected. It seems to me this is precisely the situation at issue in the Carter reasoning. Now, the question is, if we look at the species that this individual was chosen from, it turns out they were individual number m from that population. The question is now, what is the probability that their species ever gave birth to more than 10*m individuals?

The Carter argument claims that probability is 10%. Hence, it must follow that if we repeat this experiment a billion times, in only a billion/10 times will n > 10*m. Unfortunately, that is a false conclusion, because the actual answer depends on the distribution N(n), no assumptions about which have been specified in the usual setup of the problem (essentially because nothing is known about that distribution). For example, if we take a simple example where N(n) corresponds to a 50% chance of n=10¹¹, and a 50% chance of n=10¹⁵, then any individual with m < 10¹¹ does indeed have a 50% chance of belonging to either population (as in Grand Marquis' scenario, made more concrete here). Ergo, for that N(m) distribution, when m=10¹⁰ (as it does for us), it is false that there is a 10% chance that n > 10¹¹-- the chance is actually 50% for that m value.

Now, it could certainly be argued that in the distribution I am talking about, it is extremely improbable that m=10¹⁰ will be chosen. Nevertheless, in the spirit of "someone has to win the lottery", the key point is you cannot ask those people to make a standard probability argument-- they are in a special place. So in that is that actual N(n) in our universe, then we know we have m=10¹⁰, so we have no way at all of knowing that we are not special-- if that is really the N(n), then the m=10¹⁰ individuals are making exactly the Carter argument, and getting exactly the wrong answer.

The way I sum this up is, you cannot make any probability argument that invokes both the number m and the average n in the same argument-- because if you don't know N(n), you will be reaching a false conclusion if you do that. You can use the average n-- you can say that before you choose m, the chances are that m will not be too different from the average n (but that requires knowing what the average value of n is). Also, if the statistician did not know N(n), and was selecting m value to suss it out, that's also fine-- the first m chosen begins to give the statistician a sense of what the average n is. But nevertheless, when all is said and done, when you bin all the m=10¹⁰ individuals together, which is the bin that we are in, you find that 50% of them live in a species that will vastly outlive 10*m.

This binning proves that the Carter argument is simply wrong probability, even if you make the very same assumptions it does. Any group that uses their own m value in their calculations of expectation values is specifying something about themselves that invalidates any claim they would otherwise have of being "generic". This is quite a common error in probability that comes up in a lot of puzzles, some on this forum.