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Comment author: mwengler 23 April 2014 08:41:39PM 0 points [-]

Is it safe to say that this problem, this result, has no applicability to any similar problem involving a merely finite amount of prisoners, say a mere googol of them?

Comment author: JoshuaZ 24 April 2014 04:05:49AM 0 points [-]

Yes. But I do think that thinking critically about the assumptions you are making, in particular that you can meaningfully talk about what it means to pick a random individual in a uniform fashion, is worthwhile for understanding a fair bit of probability and related issues which are relevant in broader in contexts.

Comment author: mwengler 09 April 2014 10:42:59PM *  0 points [-]

Maybe this way then. I set down somewhere in the universe in a location that I don't reveal to you ahead of time and then identify the 100 prisoners that are closest to my position. If the 100th and 101st furthest prisoners from me are exactly equally distant form me I set down somewhere else in the universe and I keep moving until I find a location where I can identify the 100 closest prisoners to my current position.

Of that 100 prisoners, I count the number of prisoners who identified their hat color correctly.

My question is what is the probability that I have counted 50 or fewer correct answers? Is it greater than, less than, or equal to the probability that I have counted 51 or more correct answers?

Thanks to you and JoshuaZ for trying to help me here.

Comment author: JoshuaZ 23 April 2014 03:29:16AM 0 points [-]

Maybe this way then. I set down somewhere in the universe in a location that I don't reveal to you ahead of time and then identify the 100 prisoners that are closest to my position. If the 100th and 101st furthest prisoners from me are exactly equally distant form me I set down somewhere else in the universe and I keep moving until I find a location where I can identify the 100 closest prisoners to my current position.

So how have you set up the prisoners in the universe in advance and how do you decide on the location you set down?

Comment author: mwengler 08 April 2014 08:57:40PM 0 points [-]

What do you mean by half the prisoners? Let's start there.

How about I choose a prisoner at random from among all the prisoners in the problem. What is the probability that the prisoner I have chosen has correctly stated the color of the hat on his head? In particular, is that probability more than, less than, or equal to 0.5?

While we are in the neighborhood, if there is a prisoner who is more likely to get the answer correctly than not, if you could tell me what ihis step by step process of forming his answer is, in detail similar to "if he is prisoner n, he guesses his hat color is the opposite of that of prisoner n^2+1" or some such recipe that a Turing machine or a non-mathematician human could follow.

Thanks in advance

Comment author: JoshuaZ 09 April 2014 11:18:53AM 0 points [-]

How about I choose a prisoner at random from among all the prisoners in the problem. What is the probability that the prisoner I have chosen has correctly stated the color of the hat on his head?

So what do you mean to choose a prisoner at random when one has infinitely many prisoners?

Comment author: mwengler 01 April 2014 09:08:06PM -2 points [-]

In the winning strategy, do fewer than half the prisoners guess wrong? Do more prisoners guess correctly than incorrectly? I'm trying to get a handle on whether it is worth my while to try to penetrate the jargon in the "correct solutions."

Comment author: JoshuaZ 05 April 2014 10:50:27PM *  1 point [-]

What do you mean by half the prisoners? Let's start there.

Comment author: TheAncientGeek 11 March 2014 07:59:04PM 1 point [-]

Maybe the easy answer is to turn "contrarian" into a two place predicate.

Comment author: JoshuaZ 17 March 2014 01:13:58AM 0 points [-]

What would the two places be?

Comment author: Stefan_Schubert 14 March 2014 12:53:37PM *  3 points [-]

One notable fact in this regard is that even though the population of Israel is nearly half Ashkenazi jews, the most intelligent ethnic group on Earth, scoring .5 to 1 standard deviation above Europeans, Israel is not incredibly rich but actually poorer than many European countriespercapita). Even though Israel of course is a rather special case - e.g. it has had to fight a number of wars - this casts some doubt on the notion that one can infer from the fact that intelligent people in the US are generally wealthy to the notion that intelligent people are the ones who create wealth.

I do think, though, that intelligent people do in fact contribute enormously to progress and wealth (a related argument to this effect by me can be found here). However, I don't think that wealth is a very strong proxy for productivity or contribution to progress. Consider John von Neumann, for instance, who contributed spectacularly both to science, to the American war efforts in the 2nd world war and the cold war, and to the economy via, for instance, his contributions to the development of the computer. If people would have been awarded in accordance with their contributions, he would have died one of the wealthiest men on Earth.

That said, I do think that people who contribute more generally are better paid, but the relationship is not as strong as some people seem to believe. The economists' notion of "productivity" according to which higher-earning people are by definition more productive is highly misleading in this regard, since people tend to conflate this notion of productivity with the ordinary language notion of a productive person as someone who contributes a lot to wealth-creation.

Comment author: JoshuaZ 14 March 2014 01:43:59PM *  1 point [-]

There are other complicating factors in the Israel example. About 10% of Israel is ultra-orthodox (charedi) (source) and a large fraction of them are on essentially perpetual welfare with the men staying inside yeshivot all day studying and not generating any economic productivity. Also note that by some intelligence metrics other ethnic groups outscore Ashkenazim (especially Han Chinese). This is however a minor quibble and your essential points seem correct.

Comment author: DaFranker 05 March 2014 12:59:29PM *  -3 points [-]

My best take on the thing is that, historically, most great physics discoveries were made by generalist, wide-branching natural philosophers. Granted, "natural philosophy" is arguably the direct ancestor of physics from which spawned the bastards of "chemistry" and "biology", but even regardless, the key point is that they were generalists and that, if we were going to solve the current problem simply by throwing more specialized physicists and gamma ray guns at it, this is not the evidence I'd expect to see.

Given historical base rates of generalists vs specialists in physics, and the ratio of Great Discoveries made by the former rather than the latter, it feels as if generalists have a net advantage in "consolidating" recent research into a Great Discovery.

I do have to agree, though, that all of them came from physicists, if not necessarily formally trained, although in most cases they were. Good knowledge of physics is necessary, that I won't argue. But what I'll point out is that I've personally met many more game developers and programmers with a much better grasp of (basic) physics (i.e. first volume of Feynman's Lectures) than college physics department members, on a purely absolute count. It doesn't seem that far-fetched, to me, to assume there's a comparable difference in base rates of people within and outside physics departments with a solid enough grasp of physics for the Next Great Discovery, whatever that threshold may be (and obviously, the lower the actual threshold, the more likely it is that it will come from outside Physics Departments).

Comment author: JoshuaZ 11 March 2014 03:49:26PM 0 points [-]

To expand on shminux's point about what has happened in the last 100 years that's different: There's a serious lack of low-hanging fruit. Ideas are more complicated and the simple ideas that a generalist has any chance to find have to a large extent already been discovered. Note also that in fact it is well before 100 years ago that this trend already started. Darwin, Maxwell, Faraday and many other 19th century researchers were already specialists by most notions of the term. So really this trend has been going on for almost 200 years.

Comment author: JoshuaZ 04 March 2014 05:57:17AM 7 points [-]

A new paper by Lenny Susskind discusses the black hole firewall problem and suggests that the computations necessary to actually create the standard paradoxical situation are computationally intractable. Paper here, discussion by Scott Aaronson here.

Comment author: JoshuaZ 08 January 2014 12:48:58AM 2 points [-]

A new paper gives a much better algorithm for approximating max flow in undirected graphs. Paper is here. Article for general readers is here. Although the new algorithm is asymptotically better, it remains to be seen if it is substantially better in the practical range. However, this is an example of discovering a substantially more efficient algorithm where one might not have guessed that substantial improvements were possible.

Comment author: Kaj_Sotala 03 January 2014 01:49:09PM 8 points [-]

For instance, I often greet people with "How do you do?". Most people of my generation don't really know how to react to this, and it makes them stop, think, and give a more "real" answer than if I asked "What's up?" or "How's it going?".

This might backfire, though - at least in our English class, we were taught that the only acceptable response to being asked "How do you do" is to repeat "How do you do" back.

Comment author: JoshuaZ 03 January 2014 06:25:50PM 4 points [-]

I hope you didn't take that instruction too strictly or did you have another protocol for getting out of apparent infinite loops?

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