Sunday, 19 June 2016

On the Odds that an American Muslim is a Terrorist

Hi Mr. Trump,

I wrote on Twitter that someone needed to teach you about Bayes' Rule so you could understand why your idea of profiling Muslims is stupid. It is very obvious from your public statements that thinking rationally is not your strong suit, but brew yourself a cup of coffee (I know you don't drink it, but I think you should take help anywhere you can get it) and see if you might be able to follow along here, buddy.

Bayes' Rule is a pretty simple rule in probability. It's not an opinion, Mr. Trump: it's a fact. I know the difference between opinions and facts is not obvious to you. Let me explain. Occasionally there are ideas in the world that are definitely true, no matter what anyone's feelings about them may be. We call these true ideas facts. I could show you a mathematical proof of Bayes' Rule, but I don't want to strain you too hard, as I know this is probably your first foray into rational thinking. 

So please just take it as given, that this equation (Bayes' Rule) is a true fact (you can get 'the deets' here if you like):

P(A|B) = P(B|A)P(A)/P(B)

What does that mean? Well, P(A|B) is the probability of some event A being true, given that some other event B is true. Bayes' Rule says that we can figure out the (actual, true, mathematically-guaranteed) probability of P(A|B) if you can get the values for some other probabilities (that for various reasons are often easier to get): namely, the probability of event B given event A [= P(B|A)], and the individual probabilities of event A [= P(A)] and event B [= (P(B))].

It all seems very abstract, I am sure. I know that abstract thinking is difficult for you. (Indeed, it often seems like even concrete thinking is difficult for you, as you seem to flip-flop about a bewildering number of highly concrete issues.) Let's make this concrete using an example close to your heart: Let's figure out the probability that an American is a terrorist, given that they are Muslim. If that probability is high, then your idea of profiling Muslims is a good idea. If that probability is low, then your idea is not a good idea.

Bayes' rule tell us: 

P(An American is a terrorist | An American is Muslim)  
= P(An American is Muslim | An American is a terrorist) * P(An American is a terrorist) / P(An American is a Muslim)
Maybe the easiest one to start with is P(An American is a Muslim). Wikipedia says that "According to a new estimate in 2016, there are 3.3 million Muslims living in the United States, about 1% of the total U.S. population." So we have our first probability: P(An American is a Muslim) = 1% or (same thing represented a different way so you won't get confused later when we do some math) 0.01.

What about P(An American is a terrorist)? This one is a little more difficult, because it seems that you think that there are millions and millions of terrorists in the USA right now: every Mexican, every Muslim, every Canadian, the President of the United States, and so on. TechDirt had an article on this a few years ago. They estimated that there are a maximum of 184,000 terrorists in the entire world (which they also called "a ridiculously inflated level"). It's a little harder to know how many of them live in the USA because they are all hiding, biding their time. But we can estimate it roughly by looking at the proportion of terrorist deaths that occur in the USA. It has been estimated that in 2014 (a bad year for terrorism, as you may recall) there was 17,891 deaths worldwide from terrorist attacks, of whom 19 were American. Well, as you know, we had 50 deaths just a few weeks ago in Orlando, so maybe terrorism is getting worse rapidly, as you like to suggest. Let's assume that the 2014 estimate is fully ten times too low and use 190 American deaths due to terrorism per year. If deaths due to terrorism are distributed roughly proportionally to terrorists, then we can estimate that 190/17891 or about 1% of all terrorists are American. 1% of 184,000 is 1840. So now we can get what is surely a very high upper estimate on P(An American is a terrorist): the number of terrorists in the USA divided by the US population, or 1840/318.9 million, which works out to 0.0000058.

Now we only have one number left: P(An American is Muslim | An American is a terrorist). You probably disagree with most people on the planet about this number, because I know you labor under the delusion that all terrorists (including Mr. Obama) are Muslims. Researchers from Princeton University used FBI data to actually estimate this number a few years ago, and they estimated that only 10% of terrorists active in the USA are Muslim (though the estimate for the longer time period of 1970 to 2012 is much lower, just 2.5%). We will go with the larger number: P(An American is Muslim | An American is a terrorist) = 10% or 0.10.

Now we are almost done! All we have to do is plug in our numbers: 

P(An American is a terrorist | An American is Muslim) 
= P(An American is Muslim | An American is a terrorist) * P(An American is a terrorist) / P(An American is a Muslim) 
= 0.10 * 0.0000058 / 0.01 
= 0.000058
This is about 6 per 100,000, or (said another way) there is at most a 6/100,000 chance that that a random American Muslim is a terrorist. If your profilers spent just one hour profiling each random American Muslim, they will have to pass on average about 100000/6 = 16,666 hours before they profile just one terrorist. Assuming a 40 hour work week and 50 weeks of work per year, that is 416.6 person weeks or 8.33 person years per terrorist profiled. This does not strike me as a good use of resources, especially given the fact that such total concentration in identifying only Muslim American terrorists will cause you to miss the 90% of American terrorists that are not Muslim.

Let me know if you have any questions.

No comments:

Post a Comment