Big Pickles

Pi Approximation Day.

Labels: comics, math

Clark pointed out in my last post on the Washington Post's statistical analysis of the Iranian election results that we can't conclude from an event with 4 in 1000 probability of occurrence that there is a 996 in 1000 probability that occurrence of the event was faked. This is true, of course, and I should have been more clear in my remarks.

The information we want is the probability that the Iranian election results were legitimate, given the data we see. What the WaPo has given us, however, is the probability of seeing the data that we see, given a legitimate election. They sound like the same thing, but they're not.

The Wikipedia article on conditional probability gives the classic example of why this is the case. Suppose you have a test for a disease that is 99% accurate--it returns a false result 1% of the time. That sounds pretty good, but if you use it to screen for a disease that affects only 1% of the population, then the probability that a person actually has the disease, given a positive test result, drops to 50%. So the probability of seeing a positive test result, given the presence of the disease (99%) is not the same as the probability of having the disease, given a positive test result (50%).

The Prosecutor's Fallacy describes the application of this logic in the real world. Here, the minuscule probability of certain evidence emerging, given the innocence of the accused, is used to argue that the probability of the accused being innocent must be just as tiny. This is what the WaPo appears to be engaging in with their statistical analysis.

Or is it? The thing to remember here is that this discrepancy exists because of the discrepancies in the prior probabilities that our conditional probabilities are based on. For example, in our disease scenario, the test becomes as accurate as it appears if 50% of the population are affected by the disease. The probability of seeing a positive test result, given the presence of the disease, and the probability of having the disease, given a positive test result are both 99%. It's the fact that the disease is so rare to begin with that makes all the difference.

So with the Iranian election, if we already knew that there was a 50% chance the election results were faked, then the analysis described in the WaPo would be strong evidence indicating shenanigans. I'm not saying we know that, but there does seem to be a lot more evidence of irregularities other than statistical analysis alone.

One more example to illustrate what I mean. Suppose 99 out of every 100 swans are white, and one is black. If you're walking along in the park and you see a black swan, you've just witnessed a rare event, with a probability of 1%. But this does not mean that you can be 99% sure that this swan is fake. That's because the odds of seeing a fake swan are based on something else entirely, and are usually far from 50%.

Suppose, however, that you heard on the news this morning that people have been out spray-painting white swans black. Suppose further that half of them have been spray-painted. Now, when you see a black swan, the odds of it being fake have climbed much closer to 99%, because you know the odds of seeing a fake swan to begin with are 50%.

Labels: math, politics

Hello, loyal readers. My apologies for not blogging in over three months.

My excuse is that I got a new job, and this new job eats up my time in several ways. The commute is very long and the work itself gives me little opportunity to surf the Internet. So I have less time for actual blogging, and less time to find interesting things to actually blog about.

But hopefully as I settle in I can find a way to work this blog back into my routine.

To that end, check out this statistical analysis of the election results in Iran. Just by looking at the various vote tallies, it can be shown that there is a 99.5% chance that these tallies were made up by a human rather than generated randomly, as would be the case in a clean election. Fascinating.

Labels: math, politics

I never thought of this before, but it's true. Every base is base 10, in that base.

Labels: math

Let's hope this one doesn't become bimodal.

Labels: economics, math

This is similar to the idea that an infinite number of monkeys at typewriters will eventually type all of Shakespeare's plays.

In theory, the number pi, because it "goes on" forever, contains all possible combinations of digits. So any string of numbers you can conceive of is in the digits of pi somewhere.

All information that can be represented digitally takes the form of ones and zeros, or binary. So if you convert pi to binary, then conceivably you are in the possession of all information. Including a lot of stuff that could get you into trouble.

Labels: math

This blog post fascinates me.

Here's a guy who doesn't want to bother calculating the tip on his restaurant check, even though he probably has a palm pilot or laptop or cell phone out during dinner (judging by the importance of wifi access) and could easily use the calculator in said device.

So he rounds everything to even dollars, only to realize that he could be using those precious cents digits to communicate a wealth of information to himself about each meal. Simple addition, subtraction, and multiplication are too much to bother with, but binary/decimal conversions are de rigeur, apparently.

No less remarkable is his discovery that restaurants seem to be pretty lax about the total you write on your receipt.

Labels: math

This opinion piece was written at the U of U a few years ago, so maybe some of you Utah people have seen it. But it was new to me.

Basically, the author argues that pi should be the ratio of radius to circumference, rather than diameter, because that would make all the formulas look prettier. I have to agree that, once you get beyond elementary geometry, it's all radius, all the time. So why is this famous number based on diameter?

Labels: math

I saw this a while ago, but came across it again today. It's an attempt to find some unique fact about every number between 0 and 10,000.

Labels: math

If you haven't been following the recent kerfuffle over Verizon's math-challenged rates, catch up on it here.

Don't miss xkcd's response.

Labels: math