Literally every group had at least one member who supplied a P(God) of 0 and a P(God) of 100.

Okay, I'll bite: What does someone mean when they say they are Atheist, and they think P(God) = 100% ?

I would expect for asexuals to be overrepresented

Why do you expect this? It seems reasonable if I think in terms of stereotypes. Also, I guess LWers might be more likely to recognize that they are asexual.

I remember answering the computer games question and at first feeling like I knew the answer. Then I realized the feeling I was having was that I had a better shot at the question than the average person that I knew, not that I knew the answer with high confidence. Once I mentally counted up all the games that I thought might be it, then considered all the games I probably hadn't even thought of (of which Minecraft was one), I realized I had no idea what the right answer was and put something like 5% confidence in The Sims 3 (which at least is a top ten game). But the point is that I think I almost didn't catch my mistake before it was too late, and this kind of error may be common.

The first way is that the water doesn't flow heat into the beer, rather it does some work on it.

This actually clears things up quite a lot. I think my discomfort with this description is mainly aesthetic. Thank you for being patient.

We don't lose those things.

Suppose that you boil some water in a pot. You take the pot off the stove, and then take a can of beer out of the cooler (which is filled with ice) and put it in the water. The place where you're confusing your friends by putting cans of beer in pots of hot water is by the ocean, so when you read the thermometer that's in the water, it reads 373 K. The can of beer, which was in equilibrium with the ice at a measured 273 K, had some bits of ice stuck to it when you put it in. They melt. Next, you pull out your fancy laser-doppler-shift-based water molecule momentum spread measurer. The result jives with 373 K liquid water. After a short time, you read the thermometer as 360 K (the control pot with no beer reads 371 K). There is no ice left in the pot. You take out the beer, open it, and measure it's temperature to be 293 K and its momentum width to be smaller than that of the boiling water.

What we observed was:

• Heat flowed from 373 K water to 273 K beer
• The momentum distribution is wider for water at 373 K than at 293 K
• Ice placed in 373 K water melts
• Our thermometer reads 373 K for boiling water and 273 K for water-ice equilibrium

Now, suppose we do exactly the same thing, but just after putting the beer in the water, Omega tells us the state of every water molecule in the pot, but not the beer. Now we know the temperature of the water is exactly 0 K. We still anticipate the same outcome (perhaps more precisely), and observe the same outcome for all of our measurements, but we describe it differently:

• Heat flowed from 0 K water to 273 K beer
• The momentum distribution is wider for water at 0 K (or recently at 0 K) than at 293 K
• Ice placed in 0 K water melts
• Our thermometer reads 373 K for water boiling at 0 K, and 273 K for water-ice equilibrium

So the only difference is in the map, not the territory, and it seems to be only in how we're labeling the map, since we anticipate the same outcome using the same model (assuming you didn't use the specific molecular states in your prediction).

Remember, this isn't my definition. This is the actual definition of temperature used by statistical physicists.

I agree that temperature should be defined so that 1/T = dS/dE . This is the definition that, as far as I can tell, all physicists use. But nearly every result that uses temperature is derived using the assumption that all microstates are equally probable (your second law example being the only exception that I am aware of). In fact, this is often given as a fundamental assumption of statistical mechanics, and I think this is what makes the "glass of water at absolute zero" comment confusing. (Moreover, many physicists, such as plasma physicists, will often say that the temperature is not well-defined unless certain statistical conditions are met, like the energy and momentum distributions having the correct form, or the system being locally in thermal equilibrium with itself.)

I'm having trouble with brevity here, but what I'm getting at is that if you want to show that we can drop the fundamental postulate of statistical mechanics, and still recover the second law of thermodynamics, then I'm, happy to call it a feature rather than a bug. But it seems like bringing in temperature confuses the issue rather than clarifying it.

your final answer isn't right

You're right. That should be ε, not E. I did the extra few steps to substitute α = E/(Nε) back in, and solve for E, to recover DanielFilan's (corrected) result:

E = Nε / (exp(ε/T) + 1)

I used S = log[N choose M], where M is the number of excited particles (so M = αN). Then I used Stirling's approximation as you suggested, and differentiated with respect to α.

That's not an explanation, just a symptom of the problem.

This is what I was trying to convey when I said it might be another example of the problem.

I think it's reasonable, in many contexts, to say that achieving 75% of the highest possible score on an exam should earn you what most people think of as a C grade (that is, good enough to proceed with the next part of your education, but not good enough to be competitive).

I would say that games are different. There is not, as far as I know, a quantitative rubric for scoring a game. A 6/10 rating on a game does not indicate that the game meets 60% of the requirements for a perfect game. It really just means that it's similar in quality to other games that have received the same score, and usually a 6/10 game is pretty lousy. I found a histogram of scores on metacritic:

http://www.giantbomb.com/profile/dry_carton/blog/metacritic-score-distribution-graphs/82409/

The peak of the distributions seems to be around 80%, while I'd eyeball the median to be around 70-75%. There is a long tail of bad games. You may be right that this distribution does, in some sense, reflect the actual distribution of game quality. My complaint is that this scoring system is good at resolving bad games from truly awful games from comically terrible games, but it is bad at resolving a good game from a mediocre game.

What I think it should be is a percentile-based score, like Lumifer describes:

Consider this example: I come up to you and ask "So, how was the movie?". You answer "I give it a 6 out of 10". Fine. I have some vague idea of what you mean. Now we wave a magic wand and bifurcate reality.

In branch 1 you then add "The distribution of my ratings follows the distribution of movie quality, savvy?" and let's say I'm sufficiently statistically savvy to understand that. But... does it help me? I don't know the distribution of movie quality. it's probably bell-shaped, maybe, but not quite normal if only because it has to be bounded, I have no idea if its skewed, etc.

In branch 2 you then add "The rating of 6 means I rate the movie to be in the sixth decile". Ah, that's much better. I now know that out of 10 movies that you've seen five were probably worse and three were probably better. That, to me, is a more useful piece of information.

Then again, maybe it's difficult to discern a difference in quality between a 60th percentile game and an 80th percentile game.