The Bush administration and the National Security Agency (NSA) have been secretly monitoring the email messages and phone calls of all Americans. They are doing this, they say, for our own good. To find terrorists. Many people have criticized NSA’s domestic spying as unlawful invasion of privacy, as search without search warrant, as abuse of power, as misuse of the NSA’s resources, as unConstitutional, as something the communists would do, something very unAmerican.

In addition, however, mass surveillance of an entire population cannot find terrorists. It is a probabilistic impossibility. It cannot work.

What is the probability that people are terrorists given that NSA’s mass surveillance identifies them as terrorists? If the probability is zero (p=0.00), then they certainly are not terrorists, and NSA was wasting resources and damaging the lives of innocent citizens. If the probability is one (p=1.00), then they definitely are terrorists, and NSA has saved the day. If the probability is fifty-fifty (p=0.50), that is the same as guessing the flip of a coin. The conditional probability that people are terrorists given that the NSA surveillance system says they are, that had better be very near to one (p_1.00) and very far from zero (p=0.00).

The mathematics of conditional probability were figured out by the Scottish logician Thomas Bayes. If you Google “Bayes’ Theorem”, you will get more than a million hits. Bayes’ Theorem is taught in all elementary statistics classes. Everyone at NSA certainly knows Bayes’ Theorem.

To know if mass surveillance will work, Bayes’ theorem requires three estimations:

1) The base-rate for terrorists, i.e. what proportion of the population are terrorists.

2) The accuracy rate, i.e., the probability that real terrorists will be identified by NSA;

3) The misidentification rate, i.e., the probability that innocent citizens will be misidentified by NSA as terrorists.

No matter how sophisticated and super-duper are NSA’s methods for identifying terrorists, no matter how big and fast are NSA’s computers, NSA’s accuracy rate will never be 100% and their misidentification rate will never be 0%. That fact, plus the extremely low base-rate for terrorists, means it is logically impossible for mass surveillance to be an effective way to find terrorists.

I will not put Bayes’ computational formula here. It is available in all elementary statistics books and is on the web should any readers be interested. But I will compute some conditional probabilities that people are terrorists given that NSA’s system of mass surveillance identifies them to be terrorists.

The US Census shows that there are about 300 million people living in the USA.

Suppose that there are 1,000 terrorists there as well, which is probably a high estimate. The base-rate would be 1 terrorist per 300,000 people. In percentages, that is .00033% which is way less than 1%. Suppose that NSA surveillance has an accuracy rate of .40, which means that 40% of real terrorists in the USA will be identified by NSA’s monitoring of everyone’s email and phone calls. This is probably a high estimate, considering that terrorists are doing their best to avoid detection. There is no evidence thus far that NSA has been so successful at finding terrorists. And suppose NSA’s misidentification rate is .0001, which means that .01% of innocent people will be misidentified as terrorists, at least until they are investigated, detained and interrogated. Note that .01% of the US population is 30,000 people. With these suppositions, then the probability that people are terrorists given that NSA’s system of surveillance identifies them as terrorists is only p=0.0132, which is near zero, very far from one. Ergo, NSA’s surveillance system is useless for finding terrorists.

Suppose that NSA’s system is more accurate than .40, let’s say, .70, which means that 70% of terrorists in the USA will be found by mass monitoring of phone calls and email messages. Then, by Bayes’ Theorem, the probability that a person is a terrorist if targeted by NSA is still only p=0.0228, which is near zero, far from one, and useless.

Suppose that NSA’s system is really, really, really good, really, really good, with an accuracy rate of .90, and a misidentification rate of .00001, which means that only 3,000 innocent people are misidentified as terrorists. With these suppositions, then the probability that people are terrorists given that NSA’s system of surveillance identifies them as terrorists is only p=0.2308, which is far from one and well below flipping a coin. NSA’s domestic monitoring of everyone’s email and phone calls is useless for finding terrorists.

NSA knows this. Bayes’ Theorem is elementary common knowledge. So, why does NSA spy on Americans knowing it’s not possible to find terrorists that way? Mass surveillance of the entire population is logically sensible only if there is a higher base-rate. Higher base-rates arise from two lines of thought, neither of them very nice:

1) McCarthy-type national paranoia;

2) political espionage.

The whole NSA domestic spying program will seem to work well, will seem logical and possible, if you are paranoid. Instead of presuming there are 1,000 terrorists in the USA, presume there are 1 million terrorists. Americans have gone paranoid before, for example, during the McCarthyism era of the 1950s. Imagining a million terrorists in America puts the base-rate at .00333, and now the probability that a person is a terrorist given that NSA’s system identifies them is p=.99, which is near certainty. But only if you are paranoid. If NSA’s surveillance requires a presumption of a million terrorists, and if in fact there are only 100 or only 10, then a lot of innocent people are going to be misidentified and confidently mislabeled as terrorists.

The ratio of real terrorists to innocent people in the prison camps of Guantanamo, Abu Ghraib, and Kandahar shows that the US is paranoid and is not bothered by mistaken identifications of innocent people. The ratio of real terrorists to innocent people on Bush’s no-fly lists shows that the Bush administration is not bothered by mistaken identifications of innocent Americans.

Also, mass surveillance of the entire population is logically plausible if NSA’s domestic spying is not looking for terrorists, but looking for something else, something that is not so rare as terrorists. For example, the May 19 Fox News opinion poll of 900 registered voters found that 30% dislike the Bush administration so much they want him impeached. If NSA were monitoring email and phone calls to identify pro-impeachment people, and if the accuracy rate were .90 and the error rate were .01, then the probability that people are pro-impeachment given that NSA surveillance system identified them as such, would be p=.98, which is coming close to certainty (p_1.00). Mass surveillance by NSA of all Americans’ phone calls and emails would be very effective for domestic political intelligence.

But finding a few terrorists by mass surveillance of the phone calls and email messages of 300 million Americans is mathematically impossible, and NSA certainly knows that.

FLOYD RUDMIN is Professor of Social & Community Psychology at the University of TromsÃ¸ in Norway. He can be reached at frudmin@psyk.uit.no