Roger Cohen has detected and denounced the eternal crypto-Nazism of the Trumpians with such integrity and equipoise that I’m starting to worry for his eyesight. Towards the end of Cohen’s diatribe, one finds this statistical observation:

You would not guess from Trump’s words that a Cato Institute study of refugees admitted to the United States between 1975 and 2015 found that the chance of an American being killed in a terrorist attack committed by a refugee is 1 in 3.64 billion.

The study in question, by Alex Nowrasteh (published in September 2016) found no such thing. Apparently, the columnist couldn’t be bothered to cite its conclusions properly. Cohen makes it sound as if the analyst had made an estimate of a *future* event’s probability and found it extreme low. Cohen also misunderstood the meaning of the 1975-2015 time interval: it has nothing to do with the time of the refugees’ admission.

In his work, Nowrasteh set out to answer this question: What was “the chance of an American being murdered in a terrorist attack caused by a refugee” in the span from 01/01/1975 to 12/31/2015? What he did compute, if I understand him correctly, was the *frequency* of such event per year per capita. In other words, if refugees caused three deaths via terrorist attack in that period, the frequency per year was 3/41 events, and if the average population was a little above 266 million, the frequency per year per capita was 3/(41*266,000,000), approximately equal to 1 divided by 3,64 billion.

I doubt Nowrasteh’s frequency calculations add much to a good-faith observer’s understanding of the issues involved. His first paragraph is the most informative of them all:

Foreign-born terrorists who entered the country, either as immigrants or tourists, were responsible for 88 percent (or 3,024) of the 3,432 murders caused by terrorists on U.S. soil from 1975 through the end of 2015.

I would add, as a footnote perhaps, that only one of those 3,024 Americans was killed by an illegal alien, and three more, by refugees. However, Nowrasteh’s study does not even attempt risk analysis, contrary to its title. The calculation of frequencies, misrepresented as “chances” – that is, probabilities – is only performed by the author for the shock value of the “odds” he obtains, such as “one in three billion.”

Which aren’t really odds of anything in particular. Frequency, generally speaking, does not mean chance. (Nor is past frequency necessarily a good predictor of future probability, which is more or less obvious.) Even in repeated identical trials, the outcomes of a relatively short sub-series can trick people into misestimating the underlying probability distribution. For example, a fair coin often lands a series of heads or tails that some people mistake for evidence of its bias.

In a sufficiently long series of trials, long runs of heads or tails are to be expected. See, for example, Mark Schilling‘s 1990 article, *The Longest Run of Heads*, for which he won the George Pólya award. In a later article, Prof. Schilling derived a simplified rule of thumb for the longest success run and remarked:

The run lengths given by this rule of thumb are often longer than what people expect. For example, formula (1) predicts that the longest run of heads in 200 tosses of a fair coin would have length about seven. Yet few individuals, when asked to write down a simulated sequence of 200 coin tosses, include a run as long as seven consecutive heads or tails… This may go a long way toward explaining the so called “hot hand” phenomenon… in which a casual observer of a sporting contest or similar situation ascribes a long run to psychological “momentum” when it is entirely compatible with natural variation.

Needless to say, we’re not even dealing with repeated experiments here, unless we imagine identical worlds, as many as of them as we wish… see the Sunrise problem for more.

Below the cut, a more technical explanation of my understanding of Nowrasteh’s calculations.

Suppose there are *K* categories of terrorists, possibly intersecting. Suppose is the total number of “terrorist deaths” caused in the US by foreign nationals of the *k*-th type (*k* is an integer from 1 to *K*) during a period of *N* years. The US population in the *n*-th year (*n* is an integer from 1 to *N*) is . The average population for the period is the arithmetic mean, .

I understand that Nowrasteh obtains “the average chance of dying in an attack by a foreign-born terrorist” of type *k,* on U.S. soil, per year, as

.

Nowrasteh is looking at the period from the start of 1975 to the end of 2015, so *N* = 41. He appears to be using the average US population of 266 million; my calculation from the Wikipedia table gives 267 million, which isn’t much of a difference. He is using ten categories of foreign terrorists, including “All.” For example, only one American was killed in a terror attack by an illegal immigrant (line 9, Table 1) from 1975 to 2015.

Thus, if I understand Nowrasteh’s reasoning correctly, the annual chance (actually, frequency) of dying from this cause is equal to one divided by the number of years (41) and by the average population (266 million). Unsurprisingly, this probability comes to one in almost 11 billion. The total number of people killed by alien terrorists was 3,024, so the chance understandably goes up from one in 11 billion to one in 3.6 million.