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Another example of this is the case of video surveillance in airports and at sporting events, or even on the street, in the hope that we can match a face with the face of a known terrorist on file in a security database.

Your favorite homeland security director assures you that, among the general population, we expect to see one terrorist in about 1 million people (probably a very high estimate). So, the probability of any randomly chosen person being a terrorist is about 1/1,000,000, or 0.000001 (0.0001%).

Extensive trials have shown that the reliability of the video camera's ability to identify a terrorist is 99% (an extremely unrealistically high figure). More precisely, while the equipment does not fail to identify a terrorist when one is present (highly unlikely), it gives a positive result in 1% of the cases where the person is not a terrorist -- what is known as a "false positive."

When the surveillance equipment is deployed, it produces a positive identification.

What is the probability that a real terrorist has been identified?

What will be the ratio of false positives to correct identifications?

Answer

This time, P(H) is the probability that the hypothesis is correct in the absence of any evidence is 0.000001.

P(H | E) is still 1

P(H_wrong) is now (1 - 0.000001), or 0.999999 .

P(E | H) is now .01 (1%)

So, P(H | E) = 0.000001 x 1 / (0.000001 x 1 + 0.999999 x .01)

                 = 0.000001 / (0.000001 + 0.00999999)

                 = 0.000001 / 0.01000099

                 = 0.00009999

or about 0.01 % !

Example source: http://www.maa.org/devlin/devlin_2_00.html