This story has received a lot of coverage in the last few days, even extending to Jay Leno's monologue. Understandably, there is a lot of concern about the civil liberties implications. But there is also another question, answered by Corey Chivers: How likely is the NSA PRISM program to catch a terrorist?

We don't really know anything about how PRISM works (NSA =

**N**ever

**S**ay

**A**nything), but with some plausible assumptions we can estimate the answer. Suppose

- If a terrorist is in the system, the probability is 99% that PRISM will flag him/her.
- An good guy has only a 1/100 chance of being flagged as potential terrorist.
- The actual number of terrorists is quite small, say 1 in 1,000,000.

Using Bayes' rule, Chivers shows that only 1 in 10,102 of the people flagged as suspects will actually be a terrorist!

The key point is that assumption (3) completely dominates the analysis. Unless the tests are

*very*accurate, the terrorists will be thoroughly hidden in the false positives.

Actually, the problem is simple enough that you do not need to explicitly use Bayes' Theorem:

Suppose you have a population of 100,000,000 people. Then by assumption (3) you can expect 100 bad guys and 100,000,000 - 100 = 99,999,900 good guys.

The test is 99% accurate at detecting the bad guys so it will flag 99 of them.

The test has a 1% probability of flagging an innocent person, so 0.01 x 99,999,900 = 999,999 good guys will be flagged as suspects.

Hence 99 + 999,999 = 1000098 people will be flagged as suspects.

Of those suspects only 99/1000098 = 9.89902989507028E-0058 = 1/1