How many balls in the box?-meaning?

[variant 13]

Hi I read this sci-fi book a year or so ago and want to get some other peoples perspective on it.

so here it is:

1. There is a box in front of you.

2. It contains either 10 or 10000 Balls (you can't tell by the size of the box which is more likely because it could easily contain either) and you don't know.

3. One ball has your name written on it.

4. You have to press a button on the side until the ball with your name on it comes out.

5. It come out on the third press.

What does that mean? What could that mean if the balls were people?

Sorry if this is some thought experiment that has a name and stuff but I have no clue.

[Z500]

It means billions upon billions of years of countless interactions in the universe have arranged all matter so that a lump of hydrocarbons is pushing a button, and so that the one with a group of letters produced by yet other interactions on it comes out on the third push of the button.

[variant 13]

Yes maybe I should have just explained from the start:

In the book this experiment is used to illustrate that humans won't be around for much longer. Because if humans were to survive and expand out and multiple then chances are you would be born then (when there are so many more people). But your not your here, so its close to the end.

I can see a flaw in the logic, in that there were people around thousands of years ago who could have done the same experiment and thought the same thing.

[Ponzi]

Yeah Jmebob, that's the flaw I see as well. Without any sort of information about The population at the "end times" the experiment is meaningless. if that size is 10,000, then being number 9,960 would be close to the end; If that number is ten billion, then 9,960 is nowhere near the end. It also assumes a constantly increasing population, which is contrary to what we know about past populations.

Plus it suggests to me the idea that there is a set number of possible people, and that there's some sort of cosmic lottery that decides who's turn it is to go down to Earth and get born. That doesn't make much sense to me. There has to be an almost infinite amout of possible people, but obviously not all possibilites come to pass.

[JohannGoodflag]

Even if there were some 'set of possible humans', the experiment wouldn't mean much: you (probably? :D) can't repeat the experiment, so you can't really assess the "supply" of humans just from when you were born. Sure, it's extremely unlikely for you to be 3rd out of 10,000 --- but then, it's extremely unlikely for any one person to be 3rd out of 10,000. But in the end, some-one's ball comes out third. Without the ability to repeat the experiment and accumulate data, the single trial of "what number are you" is meaningless.

(Although, actually, the rate of growth being "constant" doesn't really mean anything: we could just rank people by time of birth, and make the birth order represent the order of the balls from the box. So, while being number 20 billion out of 40 billion in the rankings would be no guarantee of being at the half-way point in history, it would mean that there will only be as many humans born in the future as have existed in the past, and you could estimate the time remaining based on population growth.)

All of this argument supposes either determinism or an end-of-game analysis, of course, so it has a very high degree of mootness (either because there is no real chance involved, or because the experiment is over).

In summary, we can conclude that I enjoy wasting time on the internet.

JohannGoodflag

[Protoctista]

This, incidentally, is known as the Doomsday Argument (http://www.anthropic-principle.com/faq.html).

It's true that given the observation of the ball with my name on it coming out on the third press, it's much more probable that the urn contains only 10 balls. This reasoning can be applied to the number of people that will ever live to conclude that doom will strike soon, except if the "self-indication assumption" holds, meaning that possible worlds with more people in them have proportionally greater prior probability. This would cancel out the Doomsday Argument exactly, but may lead to other problems.

Here's half a book (http://www.anthropic-principle.com/book/) on it, very recommended if you want to study this in more depth.

[JohannGoodflag]

Hello Protoctista,

Clearly from the last two posts, we disagree. I would re-iterate that from only a single trial, your birth-number has no revealing properties about the number of humans which will come after you.

Let's remove one of the more artificial constraints of the problem. I propose that the following two thought-experiments more closely resemble real life.

Simple variant: you don't know that the box contains either 10 or 10,000: you only know that it contains less than 1e13 balls. What information do you obtain if your ball comes out third? What would you say the expeceted number of humans is, then?
More complex variant: let's say you have a box which you know for a fact contains a definite but really large supply of balls (say, 1e13). However, there are two different kinds of balls: a small ball which has numbers written on them, and a ball which is a little too large, and will jam up the dispensing mechanism if the machine tries to dispense it. Once the box tries to dispense one of the larger balls, no more balls will be dispensed.

Of the two kinds of balls, we will assume the only thing which determines how likely a ball is to be dispansed is the relative abundance of each kind of ball (that is, we suspend any physical reason why the smaller balls would settle down towards the dispensing mechanism, etc), so we only have to worry about the ratio of the kinds of balls present.

In this scenario, if you have a lucky streak and manage to exhaust all of the smaller balls, the next attempt to get a ball will jam the mechanism, and you get no more balls. More generally, the more smaller balls you dispense, the more likely it will be that the next ball you dispense will jam the mechanism, because the smaller balls are being removed while the supply of larger balls remains constant. If the machine jams, you won't know if it's because you exhausted the supply of smaller balls, or if you just got unlucky.

Now, if your ball comes third, what can you infer about the number of balls with names written on them? What is the expected number of smaller balls that you would predict? If the machine jams, what would be the number of small balls you would predict in total?

Yes, these problems are noticeably different from the original, but I suggest that the same problems that arise in trying to analyse these problems will arise in the original problem --- it's just that here, those problems are more obvious.

And again, the fact that this 'doomsday argument' is fallacious is suggested by Jmebob: anyone could have made the same argument, including the first Sumerians to settle in large communities. In the statement "the doomsday is near", if we assume "near" means something stronger than "oh, well, sometime in the next six milennia or so", then the argument clearly fails. Not to mention, as both Ponzi and I have pointed out, the whole premise of the argument supposes some rather arbitrary things about humanity and the universe.

JohannGoodflag

(editing: typos)

[John Page]

Originally posted by Z500
It means billions upon billions of years of countless interactions in the universe have arranged all matter so that a lump of hydrocarbons is pushing a button, and so that the one with a group of letters produced by yet other interactions on it comes out on the third push of the button.
:notworthy

[Protoctista]

Originally posted by JohannGoodflag
Simple variant: you don't know that the box contains either 10 or 10,000: you only know that it contains less than 1e13 balls. What information do you obtain if your ball comes out third? What would you say the expeceted number of humans is, then?
More complex variant:

You can calculate the shift in probability for both these cases using Bayes's Theorem, and it leads to results consistent with the Doomsday Argument.

And again, the fact that this 'doomsday argument' is fallacious is suggested by Jmebob: anyone could have made the same argument, including the first Sumerians to settle in large communities.

This concern is answered in the FAQ I linked (first question).

as both Ponzi and I have pointed out, the whole premise of the argument supposes some rather arbitrary things about humanity and the universe.

It only supposes the "Self-Sampling Assumption", which means you should pretend that you were selected randomly from all observers who will ever live. You don't actually have to believe that there is such a random process, just that you should act as if there were.

[JohannGoodflag]

I don't often work with statistical analysis, but a quick web search (http://plato.stanford.edu/entries/bayes-theorem/#1) tells me that Bayes' Law can be given by

P[E](H) = P(H) P[H](E) / P(E)

where P[E](H) is the probability of a hypothesis H being true, given evidence E. So, suppose that H is the hypothesis "there are only ten balls in the box", and E is the evidence "your ball came out third". We assume (fairly) that P[H](E) = 1/10.

My question is: what probability do we assign E, and what probability do we assign H? It seems to me that there is no information about either.

This is why I maintain a single trial is insufficient evidence. Without further data, we cannot determine P(E); and I don't even know where data about P(H) is supposed to come from, although we could always assume the distribution is uniform over whatever set of possibilities we assume to represent total lack of knowledge.

Can you state more explicitly what your Bayesian argument is?

JohannGoodflag

[Adora]

Originally posted by Jmebob
What does that mean? What could that mean if the balls were people?

It means the universe is only chaos and chance thanks to the limitation of human understanding. That is the "you don't know whether it has 10 or 1000 balls inside" part. Human limitation can only go so far. We don't have x-ray vision so we can't see inside the box to guess the number of balls and the probability factor. We don't have super-sensitive taste/smell senses like many mammals, so if we suddenly get a piece of food that in the outside seems perfect, and is rotten inside, many "Pick of the draw" metaphors and bandied about and its put down to chance.

If they were human, I'd hope I got the sexy one.

I think this is a variation on a common philosophy puzzle presented to first years. I remember getting something like it in my first year anyway. However, it was in a subject I failed at the time, and probably could ace now.

[tommyc]

Originally posted by Protoctista
It only supposes the "Self-Sampling Assumption", which means you should pretend that you were selected randomly from all observers who will ever live. You don't actually have to believe that there is such a random process, just that you should act as if there were. [/B]
I believe the book in question is "Time" by Stephen Baxter.

Anyway, onto this point, about the random process. You say here that there is no random selection process but you should act as if there were one. Why?

{edit - "were" not "was", doh!}

[tommyc]

My other query about the DD argument is that surely the conclusion has to be that any civilisation will probabalistically come to an end a short while after that point at which the doomsday argument is discovered. However, I feel that in making such a conclusion I am falling into the trap of making assumptions with information from outside the sphere of the sample, Im not quite sure how though.

Also, how far down the timeline of human existence would you have to travel before you could conclude that the "end" will not come soon? Surely even if the human race has lasted for 20 billion years, you are still relatively early in a potential life span of the species, and therefore the end will still be soon? Perhaps my liberal use of the word "soon" is at fault.

I know the DD argument has stood up to far greater tests than the questions I have posed, I just have a few problems getting my head around it.