What is your distinction between “best chance” and “most likely”?

Whichever generation you bet on is going to be against the aggregate probability of you being among the others. So the way to distinguish between them is to say which ones are against a lower aggregate, which it makes sense to call the more/most likely of the options.

I haven’t been saying this is probably the last generation; only that a random person is always most likely to be part of the last generation, compared to other individual generations.

It seems to me then that the best answer in your view to the question, “which generation are we most likely to be a part of?” is “a one other than the one you bet on”, which is really to say none of them.

I haven’t been saying this is probably the last generation — AJJ

It was what you said in your first post:

This shows how unlikely it is that we’ll ever expand out into the galaxy, since it would mean we’re all part of a tiny fraction of all humans, rather than the other huge group. Instead it stands to reason that we’re at the top of graph 2’s curve. — AJJ

Ignoring the actual dates and population figures and just assuming that it's a projection of how the human population will change throughout its history, it's more likely that 2019 is somewhere in the first 2/3rds than somewhere in the last 1/3rd. That's the proper way to approach this issue.

That’s assuming we expand out into the galaxy, which I figure would involve a huge increase in population (through the occupation of other planets) that would dwarf our population up until now.

P(T1) = 1/15
P(T2) = 2/15
P(T3) = 1/5
P(T4) = 4/15
P(T5) = 1/3

So, if we have to bet on a time then betting on T5 gives us the best odds. This is where I agree with AJJ. However, given that P(T1-T4) = 2/3, it's most likely not T5. This is where I disagree with AJJ.

And, as I said before, there's no circular reasoning here, and it's the same reasoning as used in the case of which time period we're living in (it's just "people being born at time" rather than "balls being given out at time"). — Michael

But it seems to me that it should make a difference that it's possible that not all balls are in the game. The chance that there is one red ball is 100% (as we have been given a ball, so it must be at least T1). The chance that orange balls are in the game must be less than 100%, since it could still be T1. Similarly, the chance that there are yellow balls in the game must be less than 100%. Doesn't the probability that each "stage" of the game has already happened decrease linearly? It's more likely that at least 2 stages have elapsed than it is that all 5 stages have elapsed. Isn't it like the room problem? I don't know which room I am in, but I am more likely to be in room 1-4 than in room 5.

If you agree with AJJ's take on the basic probabilities, then do you also agree that we can somehow deduce the timing of humanity's demise based on just the information outlined in the thought experiment?

I’m not sure it raises the chances of being a Boltzmann brain, since for that to be likely the universe would need to have had a much longer past that it’s commonly (to my knowledge) said to have had.

And it seems to me it’s not actually possible for you (anyone) to have been anyone else, since obviously you’d not be you then. I don’t see why you being you makes it likely that everyone else is a zombie, rather than everyone else just being the particular conscious person they are. — AJJ

Is this not the same question in essence? Only instead of asking it for all humanity (or conscious beings) over time, we are asking the question for all humans currently alive.

If indeed the chance of you being you is one, then it also answers my original question. In such a case the chance of me being someone else in a different time is also zero. (Assuming that me being me is connected to me being me in the time I live in).

But it seems to me that it should make a difference that it's possible that not all balls are in the game. The chance that there is one red ball is 100% (as we have been given a ball, so it must be at least T1). The chance that orange balls are in the game must be less than 100%, since it could still be T2. Similarly, the chance that there are yellow balls in the game must be less than 100%. Doesn't the probability that each "stage" of the game has already happened decrease linearly? — Echarmion

Yes, you're right. The chance that current time T >= 1 is 1, the chance that T >= 2 is 14/15, the chance that T >= 3 is 4/5, the chance that T >= 4 is 3/5, and the chance that T >= 5 is 1/3. But that works out as exactly the same odds I gave above:

P(T = T1) = 1/15
P(T = T2) = 2/15
P(T = T3) = 1/5
P(T = T4) = 4/15
P(T = T5) = 1/3

I don't know which room I am in, but I am more likely to be in room 1-4 than in room 5. — Echarmion

I agree with that if there are more people in rooms 1 - 4 than there are in 5.

If you agree with AJJ's take on the basic probabilities, then do you also agree that we can somehow deduce the timing of humanity's demise based on just the information outlined in the thought experiment? — Echarmion

No.

I like the ball example, but as I am a simple man, the ball abstraction is the only abstraction I want to make, so let's use real numbers:

I read somewhere that in our past, 100 billion people have lived (black balls). with roughly 8 billion currently alive (green balls):

so let's say graph 1:
100 black balls
8 green balls
1 million yellow (future humans) balls

and graph 2:
100 black balls
8 green balls

Now indeed, the chance of getting a green ball is a lot higher in the case of graph 2. I think no one is denying that.

But that's not really the question.

From the perspective of our balls, every single one of them has had a life. So basically I am not removing one ball from each bag. I am removing all balls from the bag.
The green ball knows not about the yellow balls, so when I ask him the question, he will assume graph 2 is more likely (which is true). Yet graph 1 can also be true.

But in fact the question is: Is the green ball more likely to be the green ball in the second case than in the first case?

It is a question I believe not only of math, but also one of consciousness. I actually liked the answer of . Even though I don't completely understand it, I think he is thinking in the right direction here.

I don’t think so: We know who we are among everyone currently alive, but we don’t know where in human history we all are, except that we’re currently heading it.

No. — Michael

Which begs the question how we are supposed to calculate such a probability in the first place if we lack information on it.

I agree with that if there are more people in rooms 1 - 4 than there are in 5. — Michael

Ok, so this is how I think the probabilities look when we don't know what "stage" we are in. Since we don't have information on the length of the stages, we should assume we are equally likely to be in each one.

T1: 1/5
R: 1/1 -> 1/5

T2: 1/5
R: 1/3 -> 1/15
O: 2/3 -> 2/15

T3: 1/5
R: 1/6 -> 1/30
O: 2/6 -> 2/30
Y: 1/2 -> 1/10

T4: 1/5
R: 1/10 -> 1/50
O: 1/5 -> 1/25
Y: 3/10 -> 3/50
G: 2/5 ->2/25

T5: 1/6
R: 1/15 -> 1/75
O: 2/15 -> 2/75
Y: 1/5 -> 1/30
G. 4/15 -> 4/75
B: 1/3 -> 1/15

This neatly reverses the probabilities. Because red is always in the game, it has a total probability of 1/3, and we should now always bet red.

Have I made some grave error here?

Let's ignore the future for a bit.

Why would it be ok to think we would had a probability of being born (as someone else) in the past (just a high one now, hence the graphs), but not ok to have the probability to have been born as someone else in the present?

I don’t think either is OK, since we can’t be anyone else. The thought experiment involves abstracting yourself from history then putting yourself back in randomly. We find that the last generation is the most likely one for us to wind up in, so that’s what makes sense for us to think, even though it’s not unlikely we’re somewhere else.

I don’t think either is OK, since we can’t be anyone else. The thought experiment involves abstracting yourself from history then putting yourself back in randomly. We find that the last generation is the most likely one for us to wind up in, so that’s what makes sense for us to think, even though it’s not unlikely we’re somewhere else. — AJJ

That's not quite how the argument goes. If we exited history and then randomly re-entered it, we might indeed be justified in reasoning we'd enter somewhere nearer the end of humanity. But the argument had been whether or not probability theory tells us there is an increased probability we are close to the end right now.

Well it’s how the argument I’ve been making goes.

Ok, so this is how I think the probabilities look when we don't know what "stage" we are in. Since we don't have information on the length of the stages, we should assume we are equally likely to be in each one.

T1: 1/5
R: 1/1 -> 1/5 — Echarmion

That's wrong. When the experimenter is deciding who to give the red ball to at T1 there's only a 1/15 chance that he picks me. Therefore there's a 1/15 chance that when he gives me a ball it's red, and so a 1/15 chance that when he gives me a ball it's T1.

The Doomsday argument is an interesting one to consider:

Denoting by N the total number of humans who were ever or will ever be born, the Copernican principle suggests that any one human is equally likely (along with the other N − 1 humans) to find themselves at any position n of the total population N, so humans assume that our fractional position f = n/N is uniformly distributed on the interval [0, 1] prior to learning our absolute position.

f is uniformly distributed on (0, 1) even after learning of the absolute position n. That is, for example, there is a 95% chance that f is in the interval (0.05, 1), that is f > 0.05. In other words, we could assume that we could be 95% certain that we would be within the last 95% of all the humans ever to be born. If we know our absolute position n, this implies an upper bound for N obtained by rearranging n/N > 0.05 to give N < 20n.

If Leslie's figure is used, then 60 billion humans have been born so far, so it can be estimated that there is a 95% chance that the total number of humans N will be less than 20 × 60 billion = 1.2 trillion. Assuming that the world population stabilizes at 10 billion and a life expectancy of 80 years, it can be estimated that the remaining 1140 billion humans will be born in 9120 years.

One of the criticisms of the Doomsday argument can apply to your example:

The a posteriori observation that extinction level events are rare could be offered as evidence that the DA's predictions are implausible; typically, extinctions of a dominant species happens less often than once in a million years. Therefore, it is argued that human extinction is unlikely within the next ten millennia. (Another probabilistic argument, drawing a different conclusion than the DA.)

In Bayesian terms, this response to the DA says that our knowledge of history (or ability to prevent disaster) produces a prior marginal for N with a minimum value in the trillions. If N is distributed uniformly from 10¹² to 10¹³, for example, then the probability of N < 1,200 billion inferred from n = 60 billion will be extremely small. This is an equally impeccable Bayesian calculation, rejecting the Copernican principle on the grounds that we must be 'special observers' since there is no likely mechanism for humanity to go extinct within the next hundred thousand years.

Simply put, the fact that we're more likely to pick 2019 from graph 2 than from graph 1 is countered by the fact that an extinction level event in the near future isn't very likely.

Sometimes physical possibility trumps a mathematical puzzle.

Well it’s how the argument I’ve been making goes. — AJJ

In that case your conclusions on the first page:

This shows how unlikely it is that we’ll ever expand out into the galaxy, since it would mean we’re all part of a tiny fraction of all humans, rather than the other huge group. Instead it stands to reason that we’re at the top of graph 2’s curve. — AJJ

Don't follow from your argument.

That's wrong. When the experimenter is deciding who to give the red ball to at T1 there's only a 1/15 chance that he picks me. Therefore there's a 1/15 chance that when he gives me a ball it's red, and so a 1/15 chance that when he gives me a ball it's T1. — Michael

Uh, what do you mean "who to give the ball"? It was never mentioned that the balls are distributed among 15 people. This is all based on the assumption that you get one ball out of a pool of 15 balls.

Given that whether or not the other 14 balls are lying in a box somewhere or in the hands of 14 other people is irrelevant from the perspective of the "player", I also don't see how you arrive at different probabilities here. If it's T1 and you have a ball, the ball must be red. It's not possible for it to be T1 and the red ball being in some other person's hands, because you have a ball.

The Doomsday argument is an interesting one to consider: — Michael

While the argument is mathematically different, it's the same kind of statistical analysis. So the same criticism would apply: how could we generate information (in the form of a probability) about the end of humanity based on the input information?

Uh, what do you mean "who to give the ball"? It was never mentioned that the balls are distributed among 15 people. This is all based on the assumption that you get one ball out of a pool of 15 balls.

Given that whether or not the other 14 balls are lying in a box somewhere or in the hands of 14 other people is irrelevant from the perspective of the "player", I also don't see how you arrive at different probabilities here. If it's T1 and you have a ball, the ball must be red. It's not possible for it to be T1 and the red ball being in some other person's hands, because you have a ball. — Echarmion

In memory of the discussion we had a while back on the Sleeping Beauty problem, say you're going to be put to sleep. A scientist will select one of the coloured balls at random. If it's red then you'll be woken on Monday; if it's orange you'll be woken on Tuesday; and so on.

Before you're put to sleep you're asked about the probability that you will be woken on Monday. That probability is 1/15. You're put to sleep and then woken up on your randomly selected day. What's the probability that it's Monday? It's still the same 1/15 it was before you were put to sleep.

While the argument is mathematically different, it's the same kind of statistical analysis. So the same criticism would apply: how could we generate information (in the form of a probability) about the end of humanity based on the input information? — Echarmion

It's described in the argument:

"If Leslie's figure is used, then 60 billion humans have been born so far, so it can be estimated that there is a 95% chance that the total number of humans N will be less than 20 × 60 billion = 1.2 trillion. Assuming that the world population stabilizes at 10 billion and a life expectancy of 80 years, it can be estimated that the remaining 1140 billion humans will be born in 9120 years"

That's not quite how the argument goes. If we exited history and then randomly re-entered it, we might indeed be justified in reasoning we'd enter somewhere nearer the end of humanity. But the argument had been whether or not probability theory tells us there is an increased probability we are close to the end right now. — Echarmion

That's correct.

Well it’s how the argument I’ve been making goes. — AJJ

In that case I'm afraid we have had a miscommunication and have both been talking about something else :P

In memory of the discussion we had a while back on the Sleeping Beauty problem, say you're going to be put to sleep. A scientist will select one of the coloured balls at random. If it's red then you'll be woken on Monday; if it's orange you'll be woken on Tuesday; and so on.

Before you're put to sleep you're asked about the probability that you will be woken on Monday. That probability is 1/15. You're put to sleep and then woken up on your randomly selected day. What's the probability that it's Monday? It's still the same 1/15 it was before you were put to sleep. — Michael

I think I know where we are talking past each other. You are looking at this from the perspective of the experimenter, who decides in advance what ball to give to the player, knowing the sequence of colours. So from you perspective, all balls are always in the pool.

I am looking at this from the perspective from the player who just receives a ball. The ball can be drawn from a machine. The machine draws from a pool of balls that changes over time in the manner outlined above. In that case, we have to account for the fact that the machine will more likely than not have drawn from a limited pool.

What I want to illustrate is that, if the pool of balls is not fixed in advance, the probabilities can change quite drastically. In such a case, merely assuming to have drawn from the largest group is mistaken. This is the case for the argument made in this thread. There is no "experimenter" that knows the sequence in advance and has assigned you a slot in the history of humankind. You simply know you "drew" a slot, but not whether or not the pool is limited. In that situation, you need further information.

It's described in the argument:

"If Leslie's figure is used, then 60 billion humans have been born so far, so it can be estimated that there is a 95% chance that the total number of humans N will be less than 20 × 60 billion = 1.2 trillion. Assuming that the world population stabilizes at 10 billion and a life expectancy of 80 years, it can be estimated that the remaining 1140 billion humans will be born in 9120 years" — Michael

Sure, that's what happens mathematically. But since we're on a philosophy forum, let's look at it from an epistemological perspective. It's not actually possible to compute the (likely) end of humanity based on the input information here, would you agree? If so, then what does that probability actually mean?

The Doomsday argument is an interesting one to consider:

Denoting by N the total number of humans who were ever or will ever be born, the Copernican principle suggests that any one human is equally likely (along with the other N − 1 humans) to find themselves at any position n of the total population N, so humans assume that our fractional position f = n/N is uniformly distributed on the interval [0, 1] prior to learning our absolute position.

f is uniformly distributed on (0, 1) even after learning of the absolute position n. That is, for example, there is a 95% chance that f is in the interval (0.05, 1), that is f > 0.05. In other words, we could assume that we could be 95% certain that we would be within the last 95% of all the humans ever to be born. If we know our absolute position n, this implies an upper bound for N obtained by rearranging n/N > 0.05 to give N < 20n.

If Leslie's figure is used, then 60 billion humans have been born so far, so it can be estimated that there is a 95% chance that the total number of humans N will be less than 20 × 60 billion = 1.2 trillion. Assuming that the world population stabilizes at 10 billion and a life expectancy of 80 years, it can be estimated that the remaining 1140 billion humans will be born in 9120 years. — Michael

This is exactly the name I was looking for, thanks!
One thing I don't understand from the quoted text though (still have to read the wiki): Why is the conclusion that there is a 95% chance there will be still 1140 billion humans to come? As I understand it, the chance is 5% sure that there is a 95% chance of 1140 billion humans still to come.

Simply put, the fact that we're more likely to pick 2019 from graph 2 than from graph 1 is countered by the fact that an extinction level event in the near future isn't very likely.

Sometimes physical possibility trumps a mathematical puzzle. — Michael

Still, the mathematical puzzle is interesting. I think it's worth the discussion. The physical facts do not apply to the puzzle itself :)

Well, for Presentism and neo-Kantianism, the probability of living now is one :)

From the standard realist perspective, averaging over all possible futures that are consistent with current cosmological information makes the probability of living at this moment of time vanishingly small, i.e. undetermined but convergent towards zero.

Well, for Presentism and neo-Kantianism, the probability of living now is one :)

From the standard realist perspective, averaging over all possible futures that are consistent with current cosmological information makes the probability of living at this moment of time vanishingly small, i.e. under-determined but convergent towards zero. — sime

I think it's an interesting point.
The graph is discriminating towards humans. What about all other conscious beings that ever existed/will ever exist in the universe (that might ask the same question). I think they should also be plotted on the graph. ;)
Then indeed the possibility either way will probably be close to zero...

What I want to illustrate is that, if the pool of balls is not fixed in advance, the probabilities can change quite drastically. In such a case, merely assuming to have drawn from the largest group is mistaken. This is the case for the argument made in this thread. There is no "experimenter" that knows the sequence in advance and has assigned you a slot in the history of humankind. You simply know you "drew" a slot, but not whether or not the pool is limited. In that situation, you need further information. — Echarmion

It's true that we can only calculate the probabilities if we know the distribution, but the questions being posed in this discussion are in the form "if the human population over time is distributed like this then what is the probability that we will live during this time period" in which case we have a known distribution from which to calculate a probability.

Obviously in real life we don't know how the human population will be distributed. We don't know the rate at which the population will grow or decline or stabilise over time, which is why I answered that we can't know when humanity will end. The only thing I've been trying to argue is that if the human population grows at a steady rate until an immediate end such that the last generation is the largest it doesn't follow that we are probably the last generation – that's only true if the last generation contains more than half of all humans who will ever live.

Still, the mathematical puzzle is interesting. I think it's worth the discussion. The physical facts do not apply to the puzzle itself — Mind Dough

The issue with the puzzle is that we're not disembodied souls that are randomly placed in one of any of the human bodies which will ever live. I just am this particular physical body (and its emergent consciousness), and this particular physical body (and its emergent consciousness) isn't equally likely to have been born at any point in human history. It could only have been born to my parents, who in turn could only have been born to their parents, and so on. There's some degree of latitude (e.g. I was born premature, but it's possible that I could have been born late), but it's limited (e.g. I couldn't have been born in China 1,000 years ago).

The puzzle is like asking for the probability that of the countless animals that have ever lived on Earth, what is the probability that only members of the human species will have the ability to speak – and asserting that each member of every species is equally likely to have this ability. Given this assertion we would expect that most speaking animals wouldn't be human as most animals aren't human, but the ability to speak isn't something that is just randomly assigned to a species at birth, just as being born in 1988 isn't something that was just randomly assigned to me.

In that case your conclusions on the first page:

This shows how unlikely it is that we’ll ever expand out into the galaxy, since it would mean we’re all part of a tiny fraction of all humans, rather than the other huge group. Instead it stands to reason that we’re at the top of graph 2’s curve.
— AJJ

Don't follow from your argument. — Echarmion

They do though. If you’re plucked out of a history that ends with us having colonised the galaxy and put back in randomly, it’s highly unlikely you’ll wind up in this tiny segment of the total population. It’s therefore unlikely that space colonisation is going to happen.

If you’re plucked out of a history that ends with us having colonised the galaxy and put back in randomly, it’s highly unlikely you’ll wind up in this tiny segment of the total population. It’s therefore unlikely that (space colonisation) is going to happen. — AJJ

Your conclusion doesn't follow.

OK... will you deign to share why?