I cannot claim to be the first person to think of this problem.
However I am certainly one of its most interested custodians. :)

I will present it in my own words:

Suppose you have two buckets of marbles in front of you. You know one
bucket contains 10 marbles and that the other contains 1 million
marbles. The marbles in each bucket are labled 1, 2, 3, 4, etc. This
is to say that in one bucket the marbles are labled from 1 to 10,
while in the other bucket they are labled from 1 to 1,000,000.

You know all this but you don't know which bucket is which. 

You reach into a randomly chosen bucket and withdraw a marble without
looking into the bucket. The marble is number 6.

That your chosen marble is number 6 is very strong evidence that the
bucket from which you pulled the marble is the one that contains only
10 marbles. The probability that that you would choose a marble
numbered 6 from a bucket containing 1 million marbles is miniscule,
while the probability that you would choose number 6 from a bucket
containing 10 marbles is relatively high.

The actual posterior probablity that your chosen bucket contains 10
marbles is roughly 0.999, very close to 1 (certainty).

The above should be considered as fact, because it is fact.

Now then here is my question:

Consider two possible scenarios for the human species, Scenario 1 and
Scenario 2.

Scenario 1 
In this first scenario our species becomes extinct in the very near
future, such that the total number of humans ever to live is close to
the present number to have ever lived (about 60 billion).

Scenario 2
In this second scenario our species survives and goes on to colonize
other planets in other solar systems over the next many thousand
years. In this scenario the total number of humans ever to live might
be something like 20 trillian.

The two scenarios are analagous to the two buckets of marbles above.

Now you find yourself looking at your personal marble number (your
personal marble number is  your numerical place in the sequence of
human births). Because only about 60 billion humans have ever lived,
and because you are among the most recent of them, your number is a
number close to 60 billion.

Because your number is close to 60 billion and nothing remotely close
to, say, 20 trillion, the human species is extremely likely to become
extinct in the near future. This is so for the same reason that you
were extremely likely to have selected a marble from the 10 marble jar
in the first part of this message.

Therefore, contrary to our intuition and native optimism, we humans
are doomed to become extinct much sooner than we would expect.

What, if anything, is wrong with this argument?

-gts

Dear gta:

Thank you for another interesting math question.

The flaw in your argument is one of a false bifurcation. You have
assumed two choices:

#1. Humans will inevitably become extinct very soon.
#2. Humans have populated the galaxy.

However, the unproven assumption is that given there are 60 billion
people who have ever lived, we will become extinct. This assumption
has nothing to do with what number I am. You relied on the assumption
that we WILL become extinct in the near future, and only if your
assumption in indeed true, will your conclusion hold. But you will
need to prove your assumption to be true first.

If and only if it is a given assumption, will the conclusion hold. For
example, I could easily rewrite your problem this way:

Scenario 1  
In this first scenario our species becomes pink furry bunnies wearing
bikinisin the very near future, such that the total number of humans
ever to live is close to the present number to have ever lived (about
60 billion).

Now you find yourself looking at your personal marble number (your 
personal marble number is  your numerical place in the sequence of 
human births). Because only about 60 billion humans have ever lived, 
and because you are among the most recent of them, your number is a 
number close to 60 billion. 

Because your number is close to 60 billion and nothing remotely close
to, say, 20 trillion, the human species is extremely likely to become
pink furry bunnies wearing bikinis in the near future. This is so for
the same reason that you were extremely likely to have selected a
marble from the 10 marble jar in the first part of this message.

Do you see the problem here? I'm assuming that we WILL become pink
furry bunnies wearing bikinis, without proving it so. So, the scenario
is meaningless.

I hope this answers your question! If you need any clarifications,
please don't hesitate to ask.

Best Regards,
blader-ga

---

Request for Answer Clarification by gts-ga on 22 Jul 2002 22:11 PDT

Sorry blader but in this case you completely miss the point.

The human species might very well evolve into bikini-clad
bunny-rabbits. Or it might evolve into pin-striped polar bears. I
don't care. It makes no difference whatosever to my question how the
human species is defined to have ended.

> However, the unproven assumption is that given there are 
> 60 billion people who have ever lived, we will become extinct.

This is not an assumption in my problem, blader. It is possible
conclusion.

Accurately:
It is my thesis here that the human species must end relatively soon,
because my having found myself to be about the 60 billionth human is
much more likely to have happened in a universe in which only 60
billion humans will ever live than in a universe in which 20 trillian
humans will ever live.

-gts

---

Clarification of Answer by blader-ga on 22 Jul 2002 23:20 PDT

Dear gts:

Re: 
 
"It is my thesis here that the human species must end relatively soon,
because my having found myself to be about the 60 billionth human is
much more likely to have happened in a universe in which only 60
billion humans will ever live than in a universe in which 20 trillian
humans will ever live."

The Simple Explanation:

The problem with this thesis is that these two "different" universes
are really the same one, so there is only one "bucket." The original
analogy doesn't apply. Allow me to use a different analogy:

Assume we have an empty bucket, and a numbered ball is added to the
bucket every second. Let's say we know that at some random time (from
0 to infinity seconds), the balls will all disappear. We reach into
the ball, and the number we got is say the 60 billionth one. Does the
fact that we got the 6th, 60th, 600th, or 60 billionth tell us
anything about when the balls will disappear? No. Why should it?

The more technical explanation would be that you're limiting yourself
to just two choices, one is 60 billion, the other 20 trillion. To
really analyze this, we'll need to use formal statistics such as
cumulative probability functions, but let's see if I can use plain
english. Agreed, the odds that you are "almost" the 60 billionth out
of the universe with 20 trillionth possible people is relatively low.
But there are not just these two universes. There could be universes
with 21 trillion people, or 22, or 23, ad infinitum. You could have
been very likely born in any one of these infinite number of other
universes that DIDN'T go extinct, and still have been 60 billionth.

If you need more clarifications, please don't hesitate to ask. It's my
pleasure to help you further.

Best Regards,
blader-ga

---

Clarification of Answer by blader-ga on 22 Jul 2002 23:52 PDT

Dear gta-ga:

Although I stand my original answer and clarifications, I was informed
by googlebrain-ga (thank you!) that this is referred to as the
Doomsday Argument. I must admit I had never heard of this before, and
after doing research, there appears to be much debate over the
solution! Here's a great article on it:
http://www.anthropic-principle.com/preprints/inv/investigations.html

Best Regards,
blader-ga

---

Clarification of Answer by blader-ga on 23 Jul 2002 00:01 PDT

Dear gts:

After doing further research on the subject, it appears that even
after half a century, the argument is still very much alive today. I
can't imagine giving a really satisfying answer to a 5 decade old
unsolved philosophical problem, so I would completely understand if
you chose to ask for a refund.

Best Regards,
blader-ga

---

Request for Answer Clarification by gts-ga on 23 Jul 2002 09:54 PDT

blader, 

I would never think of asking you for a refund! :) 

I wanted to pose this problem purely for the sake of stimulating some
intelligent discussion, with hopes that someone here might come up
with an iron-clad refutation of my argument. I purposefully avoided
the use of the common name for the problem ("The Doomsday Argument")
because it's easy to plug it into a search engine and then regurgitate
quotations and URL's without actually giving serious thought to the
problem. Unfortunately googlebrain spilled the beans. :)

you wrote:
"Assume we have an empty bucket, and a numbered ball is added to the
bucket every second. Let's say we know that at some random time (from
0 to infinity seconds), the balls will all disappear. We reach into
the ball, and the number we got is say the 60 billionth one. Does the
fact that we got the 6th, 60th, 600th, or 60 billionth tell us
anything about when the balls will disappear? No. Why should it?"

I believe you are wrong about this. Upon withdrawing a ball you obtain
some new knowledge of the sequence. If you pulled ball number 6, for
example, then you would then know there is a relatively high
probability that the sequence ends with a low number (e.g., between 1
and 100) as opposed to a high number (e.g., between 1 million and 1
trillion).

In fact your analogy is actually another way to present the Doomsday
Argument.

That you should have a birth-order rank of only about 60 billion is
much more
likely if only about 60 billion persons will ever have lived than if
there will be many trillions of persons to ever live.

The human race is therefore doomed, at least until someone finds a
good way around this problem. :)

---

Clarification of Answer by blader-ga on 23 Jul 2002 15:32 PDT

Dear gts:

Hm, I think I know what is the problem here. You are assuming that
there is a equal probability of you being born in a universe with 20
trillion people as there is in a universe with 60 billion. However,
there is no evidence (as terra-ga pointed out) that your universe of
birth is random at all, or that there is even a choice. Maybe
metaphysically speaking, you can only be born in this universe? Here's
where we get into the really philosophical arguments. =)

Thank you very much for your generous rating!

Best Regards,
blader-ga

---

Request for Answer Clarification by gts-ga on 23 Jul 2002 18:08 PDT

blader, 

"Hm, I think I know what is the problem here. You are assuming that
there is a equal probability of you being born in a universe with 20
trillion people as there is in a universe with 60 billion."

No, I don't make that assumption.

I note that I was born among the first 60 billion of what optimists
think will be a total population numbered in the trillions. That I
should have been born into this small fraction of that possible
population is very unlikely.

This means it is very unlikely that the optimistics are correct. The
odds are with the pessimists.

"However,there is no evidence (as terra-ga pointed out) that your
universe of
birth is random at all, or that there is even a choice."

terra's argument about this made no sense to me. See my replies to him
below.

"Thank you very much for your generous rating!"

You're welcome. Thanks for tolerating me. :-)

---

Clarification of Answer by blader-ga on 23 Jul 2002 19:03 PDT

Dear gts-ga:

"I note that I was born among the first 60 billion of what optimists
think will be a total population numbered in the trillions. That I
should have been born into this small fraction of that possible
population is very unlikely."

Here's where get to the philosophy part, I think. I would disagree
that there is anything "random" about your time of birth. The
presumption is that you have an equal chance of being born at any one
other time during the course of the universe. Philosophically
speaking, this is debatable. It's like saying, well if I wasn't born
in the 60 billionth person, I could have just as likely been born the
60.0000001 billionth, or the 20 trillionth. Maybe there really is no
random selection to speak of?
 
Best Regards,
blader-ga

---

Request for Answer Clarification by gts-ga on 23 Jul 2002 20:01 PDT

blader,

"I would disagree that there is anything "random" about your time of 
birth. The presumption is that you have an equal chance of being born
at any one other time during the course of the universe."

The only presumption I make is that my obervations about my place in
the order of humanity is a random sample taken from the observations
of all humans to ever exist. From this sample I can derive statistics
about the population from which I was drawn. You can do the same,
using yourself as a sample.

If we deny that we are random samples taken from the population of all
humans ever to exist then the world becomes completely nonsensical.
Your empirical observations would not necessarily be consistent with
my empirical observations, and vice versa.

---

Request for Answer Clarification by gts-ga on 23 Jul 2002 20:13 PDT

Here is another link that explains the Doomsday Argument. This article
is, I think, a little more palatable than the article posted at the
URL that you offered earlier. It is by the same author (Nick Bostrom
at Yale University).

http://24.86.132.253/090101/feature/feature.htm

---

Clarification of Answer by blader-ga on 23 Jul 2002 20:41 PDT

Dear gts-ga:

Thanks a lot! Will be sure to read it. =)

Best Regards,
blader-ga

An 'A' for effort. :)

Actually, I discussed this problem in the context of a probability
course.  The class came to the consensus that the argument was flawed.

The problem lies in the phrase "You reach into a randomly chosen
bucket".  You skimmed over a key assumption here: what you really
meant is that you chose a marble from one of the two buckets with
*equal probability*.  (In a strict mathematical sense, "randomly"
implies nothing about the probabilities except that they aren't 0 or
1.)

Making this assumption explicit makes the argument much more dubious. 
In a situation where you pick a marble from two buckets, it is
certainly plausible that you do so with 50-50 chance.  But does this
assumption hold when discussing probability of birth?  We know
absolutely nothing about the probability distribution that decides who
you are born as.  There is no evidence to suggest that it is uniformly
distributed across all humans who ever will live.  Maybe for some
reason there was a high probability that you should be born as
yourself.  The tools of probability cannot be applied in any way on a
completely unknown distribution.

Probability has lots of subtle gotchas, I get caught by them all the
time too :).  Be very wary of seemingly plausible arguments based on
probabilities.


Hope this helps.

(By the way, I think I posted a partial comment by mistake.  Sorry.)

My issue with the problem is that it seems dependent upon the timing
of when one pulls their marble from the bucket.  Also, your two
buckets exist simultaneously as "stated fact" and can be accepted as
stated fact.  However, your two scenarios for the human species, which
you offer as parallels, are based upon projections to unknown
outcomes.

First, timing.  For your model to hold up, wouldn't it need to also
hold up if I travelled back in time to draw similar marbles from
similar buckets?  Let's say there had only been 1 million people
living up to that time, and the scenarios were between the human
species making it to 1 million people and soon going extinct or
reaching 60 billion and mastering global resources.  When I travel
back in time and pull my marble from the "1 million bucket" it would
tell me the species should go extinct soon... when in fact, we later
find that the human species went on to 60 billion.  Either, I guessed
incorrectly at which bucket I pulled from, or, the bucket model simply
had no way to correctly indicate a scenario based upon unknown
outcomes.

Second, comparing buckets of marbles to stated projections to unknown
outcomes.  If I pull a theoretical marble from the 60 billion bucket,
it may indicate the odds that I am part of a population measured in
billions rather than trillions.  However, it does not follow that the
60 billion bucket necessarily indicates future extinction.  Likewise,
had I pulled from the 20 trillions bucket, it would not necessarily
indicate that the human species had colonized other planets.

Since I have not put this into mathematical vocabulary for common
errors, I would be curious to know if my objections come close, and
what the correct vocabulary would be.  The best I could muster is that
your problem seems to based upon a spurious correlation between
pulling marbles from buckets and projecting future outcomes for
humanity.  (However, that can't be correct, since a spurious
correlation should at least appear accurate without evidence, while
the stated problem at no time appears to correlate) <grin>.

terra-ga wrote:

"The problem lies in the phrase "You reach into a randomly chosen
bucket".  You skimmed over a key assumption here: what you really
meant is that you chose a marble from one of the two buckets with
*equal probability*.  (In a strict mathematical sense, "randomly"
implies nothing about the probabilities except that they aren't 0 or
1.)"

I understand what you're saying but I do not see how it solves the
real-world problem faced by us as we prognisticate about the future of
our species. In the real problem (not the marble analogy) there is no
"randomly chosen bucket" with equal probability or otherwise.

In the real-world problem you simply find yourself to be of human
birth order rank approximately 60 billion. You then realize the
following (which may or may not be true):

If the human species is destined to be a long term success, such that
the total number of humans ever to live will be numbered in the many
hundreds of billions or trillions, then the total number of humans to
live thus far is a only a small fraction of the total number of humans
that will live. It is extremely improbable that I should have been
born among this small fraction of humans. It is therefore extremely
unlikely that humanity is destined to be a long term success.

continuing, terra-ga also wrote:
"We know absolutely nothing about the probability distribution that
decides who
you are born as.  There is no evidence to suggest that it is uniformly
distributed across all humans who ever will live. Maybe for some
reason there was a high probability that you should be born as
yourself.

This seems to be an interesting approach to the problem, but it also
seems like nonsense. What is the meaning of this statement?

"Maybe for some reason there was a high probability that you should be
born as
yourself."

??

How could I possibly have been born as someone other than myself? 

I'd say the probability of being born as oneself is very close to
1.:-)

It seems to me the bucket we have been drawn from is not the bucket
you'd need in order to make an educated guess about extinction of
humanity.  Due to our inability to pick across times (in particular in
the future), any time you pick someone's birth rank you'll get a
number someplace between 55 and 61 B.  Always.  If your picking a
certain birth rank from a bucket is to give an indication of the total
numbers of birth ranks available, your bucket should contain all time
periods, regardless of whether those time periods contain birthranks:

your bucket right now: 60 B birthranks, time period -14B b.c to 2002
a.d.
the 10 marble bucket: 60 B birthranks, time period -14B b.c to
infinity a.d.
the 10^6 marble bucket: 1,200,000 B birthranks, time period -14 b.c.
to infinity a.d.

By mere virtue of being alive today, you are stuck with bucket number
one, which is different from bucket number 2.  If you could draw from
a bucket that had an infinite time span, and you picked 60B, then you
would have good reason to be worried.  Somehow, you would have to be
able to reach through space time the same way we can reach through
space when you go to grab the marble.

my 2 cents.

Using this same logic, it is extremely unlikely that any individual is
the last person to be born.  While it is unlikely that you would reach
into the million ball bucket and pull out a ball within the range of
the ten ball bucket, it is even less likely that you would pull out
number 1,000,000.  So does this slim chance of anyone being the last
person born indicate that I should expect humanity to go on forever? 
My chances get better every day...

snapanswer-ga wrote:
"First, timing.  For your model to hold up, wouldn't it need to also
hold up if I travelled back in time to draw similar marbles from
similar buckets?"

Perhaps, but time travel is not possible in this universe. I don't
know what the laws of probability would look like in a world in which
we could travel backwards in time, but I can guess they would not be
the same laws that concern us here in this question.

We can know only what we can know here in the year 2002.

You continued: 
"If I pull a theoretical marble from the 60 billion bucket,
it may indicate the odds that I am part of a population measured in
billions rather than trillions.  However, it does not follow that the
60 billion bucket necessarily indicates future extinction."

How then do you explain the fact that your population is measured in
billions rather than trillions (keeping in mind that we are counting
all members of the population in the past, present and future).
Apparently something happened in your population to limit its size.

"Likewise,had I pulled from the 20 trillions bucket, it would not
necessarily
indicate that the human species had colonized other planets."

The question of whether or not we colonize other planets is completely
irrelevant. I mentioned it only to make the optimistic scenario easier
to conceptualize.

Since my sister already wears pink bunny slippers, as do many females,
is it not just a matter of time until we evolve into them? That way we
don't have to worry about where da heck did I put my pink bunny
slippers? Or, the cat is sleeping in them:)

Good question gts.  I wanted to add my 2c worth because I feel like
the comments haven't yet reached the heart of the matter.  There are
two
points to make.  The first has been touched on already.
(1) Using this sort of logic, every human ever born, starting with
the first caveman, would have concluded that the end of the species
was
near.  They all would have been wrong.  One may conclude that this
statistical
argument does not provide a reliable way of estimating WHEN we will
become
extinct.
(2) Now consider the ensemble of all past, present, and future humans,
and assume that each followed / follows / will follow this logic and
conclude that the end is near.  What fraction of them will get the
answer right?
We know that so far this approach has only given the wrong answer,
but under your implicit assumption that 
the population is rapidly rising, most people would get the right
answer by following your logic.  ie, if the population is rapidly
rising, then most people will be born just before the end, and so
if all people always assume the end is near, most people
will be correct.  But I don't think it's true that the population
must keep rapidly rising until a sudden mass extinction at the end.
Few species have followed that path.  Instead populations tend to 
level out at some fixed level and stay near that level for a long
time.
Suppose humans are like other species, and our population reaches
10 billion (say) and then levels out.  In this case it isn't
necessarily
true that most past/present/future people will get the
right answer with your approach.  If, eg, the population stays at
10 billion for 50 generations before a sudden cataclysm, then
only ~2% of people who assumed the end was near would have been right.
The best justification for your argument seems to be the claim
that most people who adopt it will get the right answer; but this
claim is only true if one assumes that the number of humans will
shoot up rapidly until the end.

chinaski

Thanks for your comments, chinaski.

You wrote:
"What fraction of them will get the answer right?
We know that so far this approach has only given the 
wrong answer, but under your implicit assumption that 
the population is rapidly rising, most people would 
get the right answer by following your logic.  ie, if 
the population is rapidly rising, then most people 
will be born just before the end, and so if all people 
always assume the end is near, most people will be 
correct."

Yes. This goes also to the "timing" objection posted earlier by
snapanswer. While it's true that someone in, say, 500 BC would have
been dead wrong to accept the conclusions of the doomsday argument, if
every human who has ever lived had accepted the conclusion then the
vast majority of them could still be correct.
 
"But I don't think it's true that the population 
must keep rapidly rising until a sudden mass extinction at the end."

There is no need to accept that as truth for the argument to hold. It
only matters that the population has risen rapidly in size in the
past.
 
"In this case it isn't necessarily true that most past/present/future
people will get the right answer with your approach."

We are not interested in obtaining the right answer for future people,
(even assuming there will be some).

By the way, I find it amusing and ironic (and a little frightening)
that astronomers just discovered an asteroid heading for earth, due to
hit in 2019. As far as they can tell so far, the asteriod is on a
dead-on course to earth. And it's big enough to cause our extinction,
similar to the one that probably caused the extinction of dinosaurs.

They're counting on experimental error to save us, hoping that future
observations and calculations will show that the asteroid will miss
us. It's rather strange that the future of the human species now
depends on our best scientists being *wrong*.

I have several ways to disprove this.

1.
In your analagy, you spoke of two buckets of marbles; this analogy
cannot be applied to the human species as you do, blader was on the
right track, but didn't dive it home.

I will explain using your marble analogy. Can any one marble be in two
places at the same time?  No, it is impossible.  You cannot have 2
buckets of marbles that both contain the marbles 1,2,3,4,5,6,7,8,9,10.
 Either they are in one bucket, or the other.  They cannot be in both
buckets at the same time.  If I pull a marble numbered 6 from one of
the buckets, there is 0% chance that it is in the other bucket.  Apply
this to the human species, you cannot have the same human in both
scenarios; therefore, they must be in the same group, of 1-20
trillian.  Then the odds that you will pull any number from that group
are identically the same.


2.
You give an example of a human count that has 20 trillian people in
it, using a similar rational that you did.  What are the odds that the
population of 20 trillian people WILL NOT make it to reach a count of
20 trillian and 1?  It approaches 0%.  Since you only gave a sample of
2, and this civilization is at 60 billion, the odds that it WILL NOT
reach a count of 20 trillian is significantly closer to 0% using the
same logic as you used in your question.  If you could give me a group
of 100 human civilizations, and how humans existed in it up to the
point of extinction, then we could add up all the numbers and divide
by 100 to predict the average.  A case of two will not work, as there
is too much room for error.

Thanks for the attempt, rubik, but your arguments fail...
 
You wrote:
"1... Can any one marble be in two places at the same time?  No, it is
impossible.  You cannot have 2 buckets of marbles that both contain
the marbles 1,2,3,4,5,6,7,8,9,10.Either they are in one bucket, or the
other."
  
I can change my analogy slightly to avoid that complication. Imagine
there is only one bucket in front of you, and only one set of marbles.
The bucket contains either 10 or 1,000,000 marbles. You pull a marble
and find it to be number six. This means the bucket almost certainly
contains only ten marbles.

In the same way, there is only one true complete count of the human
species. In one possible count it ends at around 60 billion. In
another possible count it is numbered in the trillions. You check your
personal count and find it to be about 60 billion. This is strong
evidence that the complete count of the human species will not be much
more than about 60 billion.

Your second argument also fails...
 
"2. You give an example of a human count that has 20 trillian people
in
it, using a similar rational that you did.  What are the odds that the
population of 20 trillian people WILL NOT make it to reach a count of
20 trillian and 1?  It approaches 0%.  Since you only gave a sample of
2, and this civilization is at 60 billion, the odds that it WILL NOT
reach a count of 20 trillian is significantly closer to 0% using the
same logic as you used in your question.  If you could give me a group
of 100 human civilizations, and how humans existed in it up to the
point of extinction, then we could add up all the numbers and divide
by 100 to predict the average.  A case of two will not work, as there
is too much room for error."

It is not necessary that we assign absolute cardinality to either of
the two scenarios. The argument holds if one assumes any two possible
scenarios such that in the first the human species dies out after
about 60 billion and in the second it goes on to any number much
higher than 60 billion.

Not that I think this thread is still open but I can't resist.

It seems to me that the problem lies in the fact that the marble
experiment is very different from the human population experiment in
one key way. A marble isn't chosen until all marbles are placed into
their respective buckets. However, the human in question is chosen
while the populations of both scenarios are still being built up. So
this is analgous to saying that you have chosen a marble numbered 6
after only the first 6 marbles have been added to each bucket. At that
point you know nothing about which bucket you have chosen because you
have no way of knowing the final totals in each bucket.

I know I'm probably leaving out some important clarifications but
seeing as how the thread's been dead for half a month or so I won't
waste any more time.  (o:

asterisk_man I think your dead on with that.  I know its been a month
but I just found this thread and I can't resist posting either.  I was
about 3/4th of the way thru it when I went and took a shower and I had
an interesting thought that might relate to this.

We are making two, posably eronios assumptions:

1) That the number of humans that have ever lived is finite.
2) That the number of humans that will have ever lived is finite.

If the number of humans that have ever lived is finite where did we
start? and how? If we can't answer these questions we have no idea
what our number even is.

Now if can answer these questions (God created adam/two aomeba decide
to have a party one day) is there anything to say that our creation
was a unique one time thing?  Supose this has all happened before?
However unlikely it was, we know it happened once.  So what are the
odds of it happening again?  Can we prove that the odds of it
happening again are zero?  If we can't prove this then even if the
human population ever reaches zero we are not assured that exstinction
has occured.

If we can't prove that it wont happen again we also can't prove that
it hasn't happened before.  And if it could have happened before we
still don't even know our number.

For all we know we have already spread out and colonized the stars. 
And may evolve into a species that never dies out.  While this may be
unlikely if it is even remotely posible then we don't know anything
about the population of this distribution.

Just something for anyone who finds this to kick around

Interesting question. Here are a few more thoughts:

(1) I think you are running into a faulty understanding of the nature
of time. If the human race ends at 61 billion people, then the
possibility of 20 trillion never existed, and the comparison makes no
sense. This ties in with what was said about the 1 millionth person,
who also would have been mistaken in thinking the race would die out.
The temporal order of the sequence has no bearing on the question of
when the sequence will end. There's no reliable information about the
possible number of humans that will ever exist, and hence, the number
that you are (say 60 billion exactly) has absolutely no bearing on the
future number that is possible. Let's try another example. Suppose I'm
the 10,000th person to have ever listened to the music group Cinerama
(formed within the last decade). 20,000 years from now, suppose 100
billion people have listened to them (ha ha). The fact that I'm 10,000
is irrelevant to predicting the future of the listeners, or the end of
the species, because the # of people possible could only have been a
small number during this time period, and I happen to exist now when
small numbers were possible. A million years from now, only larger
numbers will be possible (okay, for listening to Cinerama it may still
be a small number...let's hope not).

(2) The analogy that one would have an equal chance of being chosen
from the "bin" of 60 billion is faulty. Suppose you were born in 1970.
Say another 100 million people were born that year. You couldn't have
realistically been born prior to then, so the "bin" really contains a
much smaller number of people than you imagine. The comparison is not
between 60 billion and 20 trillion.

(3) You seem to be dazzled by the big numbers involved. Try this.
Assume there have been a billion billion ants on the planet in its
history. Thus, because an ant that starts life today is the billion
billionth, ants will soon become extinct because the this ant isn't
the billion billion billionth. Huh? Where's the causality? There has
to be a name for this statistical fallacy...but what?

How about this wrench
What if we ARE from the 20 trillion bucket
We have no way of knowing that 60 billion people existed thus far.
For all we know we came from distant galaxy, crash landed here and had
to start technology from scratch?  Its possible!