Comment author: Decius 05 November 2012 03:56:52AM 4 points [-]

How small does the chance have to be before it isn't a chance anymore?

Also, are you intending to research and vote only for the presidential election? Local offices have smaller budgets but also smaller margins...

Comment author: Mark_Eichenlaub 05 November 2012 01:00:23PM 0 points [-]

Re: other stuff on ballot. Yes, that's right. I was just replying to the content of the post.

Sorry, I don't understand what was meant by your first sentence.

Comment author: Mark_Eichenlaub 05 November 2012 02:35:04AM *  0 points [-]

I rarely make decisions involving such low probabilities, so I don't really know how to handle risk-aversion in these cases. If I'm making a choice based on a one-in-ten-million chance, I expect that even if I make many such choices in my life, I'll never get the payoff. This is quite different than handling one-in-a-hundred chances, which are small but large enough that I can expect the law of large numbers to average things out in the long term. So even if I usually subscribe to a policy of maximizing expected utility, it could still make sense to depart from that policy on issues like voting.

BTW, in my state, Maryland, Obama has a 18-point margin in the polls. That could easily be six standard deviations away from the realm where I even have a chance of making a difference.

Comment author: jsteinhardt 31 October 2012 09:12:24AM 0 points [-]

If your sample size is n times as large, the probability becomes (.05)^n

I'm not sure that follows.

Comment author: Mark_Eichenlaub 31 October 2012 12:42:26PM *  1 point [-]

You're right. That would be true if we did n independent tests, not one test with n-times the subjects.

e.g. probability of 60 or more heads in 100 tosses = .028

probability of 120 or more heads in 200 tosses = .0028

but .028^2 = .00081

Comment author: Eliezer_Yudkowsky 30 October 2012 04:45:05AM 15 points [-]

I was just mentally approximating log(.001)/log(.05) = 2.3.

Comment author: Mark_Eichenlaub 30 October 2012 03:34:28PM *  10 points [-]

Thanks. Sometimes I learn a lot from people saying fairly-obvious (in retrospect) things.

In case anyone is curious about this, I guess that Eliezer knew it instantly because each additional data point brings with it a constant amount of information. The log of a probability is the information it contains, so an event with probability .001 has 2.3 times the information of an event of probability .05.

If that's not intuitive, consider that p=.05 means that you have a .05 chance of seeing the effect by statistical fluke (assuming there's no real effect present). If your sample size is n times as large, the probability becomes (.05)^n. (Edit: see comments below) To solve

(.05)^n = .001

take logs of both sides and divide to get

n = log(.001)/log(.05)

Comment author: maia 31 July 2012 11:12:02PM 0 points [-]

By "upward," I just meant to emphasize that it was opposing gravity, e.g., positive. But of course, now that I think about it for a minute, I see that I was wrong, it is under no tension at all. Oops.

Comment author: Mark_Eichenlaub 01 August 2012 12:00:17AM *  2 points [-]

I think I see what you mean. To clarify, though, tension doesn't have a direction. In a rope, you can assign a value to the tension at each point. This means that if you cut the rope at that point, you'd have to apply that much force to both ends of the cut to hold the rope together. It's not upward or downward, though. Instead, the net force on a section of rope depends on the change in the tension from the bottom of that piece to the top. The derivative of the tension is what tells you if the net force is upward or downward. This derivative is a force per unit length.

In general, tension is a rank-two tensor, and is just a name for when the pressure is negative.

Comment author: maia 31 July 2012 10:07:19PM 3 points [-]

A way to think about this problem that clarified it for me: The top of the slinky is experiencing tension equal to its whole weight. The bottom is experiencing upward tension equal to its weight, so at the instant when it is dropped, it experiences no net force - until the upper parts of the slinky start traveling downward, removing the upward tension. (The speed-of-information-in-materials wasn't really convincing to my intuition, because it feels like, since the bottom of the slinky is in a gravitational field , it should already "know" it is falling... )

For a mantra: How about something like, "What do you know? What don't you know? How do you connect them?"

Comment author: Mark_Eichenlaub 31 July 2012 10:44:31PM *  0 points [-]

I'm not really sure what you mean by "upward tension", sorry. Tension in one dimension is just a scalar. The very bottom of the spring is under no tension at all, and the tension increases as the square root of the height for a stationary hanging slinky.

Comment author: fubarobfusco 31 July 2012 07:05:21PM 2 points [-]

Does the top of the slinky accelerate groundwards faster than gravity?

I'm not sure how to pronounce "Polya"

Tip: Use Google Translate to find pretty-good pronunciations of foreign names. Set the source language to the one the name comes from (Hungarian, in this case), type the name with the right accent marks (Pólya), and click the speaker button in the bottom right of the box.

Comment author: Mark_Eichenlaub 31 July 2012 07:13:19PM 2 points [-]

Thanks for the tip.

The center of mass of the slinky accelerates at normal gravitational acceleration. The bottom of the slinky is stationary, so to compensate the top part goes extra-fast. I did a short calculation on the time for the slinky to collapse here http://arcsecond.wordpress.com/2012/07/30/dropping-a-slinky-calculation-12/

What is the Mantra of Polya?

6 Mark_Eichenlaub 31 July 2012 05:49PM

The other day at dinner, someone showed me this video of a slinky dropping. It shoes that the bottom of the slinky stays perfectly stationary for a while after it's been dropped. (The link goes to the 10-second interesting part).

I spent some time trying to figure out why that happens, but didn't get it. The next day, I spent half an hour writing down the differential equations that describe the slinky's motion and staring at them, with no idea how to proceed. Eventually, I watched the video again with sound, and learned the simple answer, which is that the speed of waves traveling in a slinky is very slow - a few meters per second - and the bottom half sits still until a wave can travel down and inform it that the slinky's been dropped.

The strange thing is that I already knew this, or at least the idea was familiar to me. Also, while at dinner, someone mentioned the "pole-in-the-barn" paradox from special relativity, and mentioned the same speed-of-information-in-materials idea in resolving the paradox, but I still didn't make the connection to the problem I was considering.

I want a simple phrase, similar to "check consequentialism", "take the outside view", or "worth it?" that applies to checking your own thought process while solving problems to stop you from revving your engine in the wrong direction for too long. I realized I've read a book about what to do in such situations. It's George Polya's How to Solve It. (Amazon Wikipedia Google Books) I don't have a copy of the book anymore, and I would like to crowdsource creating a short phrase that captures the general mindset endorsed by it. Some questions I remember the book suggesting are

  • Have you seen a similar problem before?
  • What are the unknowns?
  • What information do you have?
  • Is it obvious that the unknowns are enough information?
There are more of these listed in the Wikipedia article.

Also, "Mantra of Polya" doesn't roll off the tongue well (at least I think it doesn't, since I'm not sure how to pronounce "Polya"), so a better name for this mnemonic would be good, too.

Comments (15)
Comment author: Mark_Eichenlaub 02 June 2012 11:52:31PM 29 points [-]

And clearly my children will never get any taller, because there is no statistically-significant difference in their height from one day to the next.

Andrew Vickers, What Is A P-Value, Anyway?

Comment author: Mark_Eichenlaub 06 May 2012 05:01:25PM *  0 points [-]

WolframAlpha is pretty good for calculating all this automatically - probably much faster than the spreadsheet. For example:

http://www.wolframalpha.com/input/?i=1+cup+milk+%2B+8+oz+yogurt+%2B+1+banana+%2B+1+slice+bread+%2B+1+slice+cheese

View more: Prev | Next

Powered by Reddit Powered by Reddit