A simple application of Bayes' Theorem will dissolve lots and lots and lots of things you fear, but irrationally. The fear du jour in light of the Boston Marathon bombing: Muslim Rage.

Undoubtedly, you probably once asked a math teacher "when am I ever gonna use this?" In light of recent developments in the investigation of Monday's bombing at the Boston Marathon, I have an answer: you'll use this when you're surrounded by demagogues calling for the waste of additional resources screening members of a minority group (if not members' outright persecution) based on the despicable actions of a small handful of people.

Here's a great post from two years or so ago in which William Easterly lampoons the statistical illiteracy of people who want to persecute or "profile" Muslims based on Muslims' role in 9/11 and other attacks and that is relevant again in light of their alleged role in the Boston bombing. The upshot: even if the probability that someone is a Muslim given that he is a terrorist is 100%—meaning that all terrorists are Muslims, which they aren't—the probability that a Muslim is a terrorist is not. In fact, the probability of a Muslim being a terrorist is, according to Easterly's calculations, 0.007% or 0.0007% (depending on which one features a typo—but even with an order-of-magnitude typo in the downward direction the probability is basically low enough to ignore).

I think a lot of people scared of Muslim Rage understand this more intuitively when we're dealing with crimes committed by white people, crimes committed with guns, and crimes committed by white people with guns. Consider school shootings. A quick Google search turned up this piece about Katherine Newman's book Rampage: The Social Roots of School Shootings. Apparently, all of the school shooters Newman studied have been white males (this article offers counterexamples, but let's just go with "all school shooters are white males" for sake of the example).

Of course, this is the kind of sophistry that gets churned out in the wake of having our stereotypes confirmed so spectacularly. Think about it: there are practically zero Chechens in the United States, but it still turns out that the most spectacular terrorist rampage in the last few years was committed by two of the 200 immigrants from a small, remote ethnic group that has still managed to develop a reputation as the World's Scariest Guys. That's stereotype rebunking at its finest.

And of course, the issue isn't just terrorism. If two regular guy Chechen brothers pulled off this, that suggests that the bell curve for the whole group is shifted in the direction of courage, violence, hostility, and a general lack of appreciation for the Golden Rule. That manifests itself in a whole lot of ways — for example, see my post below on how Chechens in Chechnya drive.

We see something similar in England. The Muslims commit the spectacular suicide bombing attacks, but that's just the far right edge of a Muslim probability distribution that is shifted overall in the direction of hostility toward the host populations, which manifests itself in rape, riot, immigration fraud, cousin marriage, welfare cheating, and petty acts of bloodymindedness.