Working in a sensitive government position, I can be expected to be asked to undergo a polygraph examination. There's a lot of talk of "fooling the lie detector" and speculation of whether or not it is accurate. I'd just like to demonstrate something statistically interesting.

For the time being, let's be generous to the polygraph and say that the probability of it correctly detecting a lie is 99% and that the rate of false positives is only 1%. This conditional probability can be written as:

P(+|L) = 0.99
P(+|T) = 0.01

Where the symbol P(a|b) indicates the probabilty of result 'a' for the subset of all cases for which 'b' is true. I'm using '+' to indicate a positive result of the polygraph, indicating that I'm lying. 'L' indicates that I am actually lying, and 'T' indicates that I am actually telling the truth. So P(+|L) = 0.99 means that there is a 99% chance that if I'm lying, the lie detector will give a positive result.

Now, let's make an assumption that is rather hostile to me - that I lie one-tenth of a percent of the time.

P(L) = 0.001
P(T) = 0.999 because P(L) + P(T) = 1

Now we apply Bayes' theorem, which states that

P(a|b)P(b) = P(b|a)P(a)

or, for our purposes, P(T|+)P(+) = P(+|T)P(T). This can be solved for P(T|+) to give:

P(T|+) = P(+|T)P(T)/P(+)

Now, to calculate P(+) we can simply make the argument that the machine will give positive results sometimes when I'm telling the truth, and sometimes when I'm lying. And the sum of those two will be the total probability of the maching giving a positive result. This can be written as:

P(+) = P(+|T)P(T) + P(+|L)P(L)

Now we have numbers for everything on the right-hand side of this equation. Plugging in our assumptions, above, we get that

P(+) = 0.01098

This is the overall probability that the lie detector will indicate that I'm lying. Now, plug this into the above equation for P(T|+) and plug our numeric assumptions into the remaining terms on the right-hand side to reveal

P(T|+) = 0.909836

The meaning of this statement is that, when the polygraph indicates I am lying, there is a 91% probability that I am actually telling the truth. Bear in mind that this result was achieved using assumptions about my lying and the accuracy of the polygraph that are extremely favorable to the polygraph examiner and hostile to me. If you assume that I lie less than 0.1% of the time and/or that the polygraph does not catch 99% of lies and/or that the polygraph has a higher than 1% false-positive rate, the P(T|+) probability goes up - quickly. For numbers that I think are more accurate, the probability of my telling the truth when they say I'm lying is more like 99.9999%.

Polygraph examiners use the results of the polygraph machine as an intimidation device - saying things like "The machine indicates that you are being dishonest. Why might that be?" My response will be:

"The machine you are using is not up to the task."