Logic question/puzzle

[Outrider]

About half a year ago, I read an article in the New Yorker which put forth some interesting theories.

The basic idea is:
1) Statistically, you are probably not special
2) Therefore, the safest bet is to always assume that you are in the average group

I know this makes no sense yet, so here are some examples:

If you open the newspaper and discover the fact that a Broadway play has been running for 4 years, chances are that it is not at the beginning of a 200 year run, nor that it will close the next day. Therefore you can assume with something like 97% accuracy that the play will probably finish its run in another 4 years, resulting in a total run of 8. However, if you check up on the same play the next day, and it is still running, that expextancy goes up a little bit. And a little bit more the next day, and so forth. Of course, you will eventually be wrong. But the total accuracy for your predictions should be good (can anyone prove/disprove this mathematically?)

Another example:

Let's assume that humanity will be destroyed in a nuclear war or by some other disaster. That means that the graph of the population of the earth vs. time results in the first half of a bell curve, followed by a sharp drop-off. Statistically, you are most likely to be living in the period where the most humans are on the planet (because you are unlikely to be "special" enough to live at the extreme ends of the curve.) In other words, if the human race will ever be suddenly destroyed, this event is statistically likely to occur in our lifetime!

I have several questions:

1) Does anyone know who proposed this concept?
2) Can anyone explain it in better detail than me?
3) Can anyone propose any flaws or strengths in this form of logic?

[carnivorousplant]

Given the definition of Bell curve, I should think that most of us would be in the middle of the curve, not at the end.

[Outrider]

To clarify: in this case, the middle of the curve represents the peak of humanity right before the huge drop-off (which represents the catastrophic event.) So it isn't really a true Bell curve, but it seems that the same principle applies. Being on the "beginning" of the curve would mean being born just as our species was emerging, whereas being on the "end" of the curve means that you are among the last humans before extinction.

[carnivorousplant]

Forgive me.
Most of the people who ever lived would be in the middle.
I, not being special, would be in the middle and save from nuclear disaster.

[APB9999]

It wouldn't be a bell curve; you're describing a curvy "sawtooth". But the reasoning as I understand it from what you said is: The probability that a randomly chosen person (from the total pool of people who ever lived) will be found to come from the last, say, hundred years is higher than that they will come from any other single hundred year period. This is simply because the population is so much larger now. So the place on the overall sawtooth curve, including future as well as past, where a randomly-chosen person is most likely to come from is the region where the population is highest: just before the crash.
As to whether you can assume that you are in this group, you have to consider not just the most probable position on the curve, but the overall probability that position represents.

To make an analogy: The MOST likely outcome of a roll of two dice is seven. It is more likely to be seven than any other SINGLE outcome. But is it more likely that a given roll will be seven than NOT seven? No, it is much more likley that a given roll will be SOMETHING other than seven.

So while it may be most likely that we (if we are randomly chosen) are near the end of the world, it is more likely that we are NOT.

As for the Broadway play, you would have to use something called Bayes' Theorem to make the kinds of statements you suggest, and yes, it acts sort of like that. I can't explain this theorem here in any reasonably clear way, but it should be in any probability book.

My instinct says that making the assumption of averageness in a case like this isn't right, but I can't explain exactly why (maybe just my emotional desire to be special! After all, my Mother said I was).

[RickG]

I read the same article. While I cannot remember the name of the man whose idea was profiled, I remember the argument to be as follows:

No instant of time is special.

Therefore, for any entity that has an indeterminate lifetime, there is an X% chance that the entity is currently in the middle X% of its lifetime.

In particular, if you know something has already existed for some length of time, there is a 95% likelihood that it has lived for somewhere between 2.5% and 97.5% of its ulitimate lifetime (the middle 95%). Alternatively, we can say that there is a 19 in 20 chance that the entity will live somewhere between 1/39 (2.5/97.5) of its current age longer, and 39 (97.5/2.5) times its current age longer.

So, if human civilization is 5000 years old, there is a 95% chance that civilization will disappear between 128 and 195000 years from now. This is a purely probabilistic argument, and has little predictive power, except in the aggregate.

According to the New Yorker article, the theorist in question had done remarkably well in predicting when Broadway plays would close based only on how long they had already been running. However, you can see that the predicted range of values is quite large.

Rick