The Premier of South Australia, Mike Rann, is embroiled in something of a sex scandal at the moment. I won’t go into all the sordid details, but, briefly, a woman by the name of Michelle Chantelois is claiming that Rann had sex with her several years ago (he was single but she was married at the time), a claim he is emphatically denying. The political vultures are circling because, given the denial, if a “smoking gun” (or perhaps more aptly a “blue dress”) is produced then Rann is cactus.
Anyway, none of this is really interesting to me. Actually, it’s hardly anybody’s business. The shenanigans these two people did, or did not, get up to in their private lives years ago is entirely between them and their families as far as I’m concerned. But today Chantelois has come out and volunteered to take a lie detector test to determine who is telling the truth. So I wonder, statistically what’s the probability that she will pass the test?
To answer this question we’ll use Bayes’ Theorem and some rather dodgy data points I picked up around the internet. It’s on the internet, you see, so it must be true.
The first bit of information we need is the probability that Chantelois is actually telling the truth. In a recent, and utterly meaningless straw poll conducted by the Adelaide Advertiser, this probability is precisely 53%.
Shortly before 3pm, 53 per cent of respondents to an AdelaideNow poll believed former Parliament House waitress Ms Chantelois was telling the truth about claims of a sexual relationship with Mr Rann, while 47 per cent believed the Premier’s rejection of the allegations.
Therefore P(Chantelois is telling the truth) = P(T) = 0.53;
and P(Chantelois is not telling the truth) = P(N) = 1-P(T) = 0.47.
The next bit of information we need concerns the reliability of polygraph tests themselves. Personally I’ve always been more than a little sceptical of the infernal things. Polygraphs smell like voodoo science to me, and according to Wikipedia,
Polygraph testing has little credibility among scientists. Despite claims of 90-95% validity by polygraph advocates, critics maintain that rather than a “test”, the method amounts to an inherently unstandardizable interrogation technique whose accuracy cannot be established. A 1997 survey of 421 psychologists estimated the test’s average accuracy at about 61%, a little better than chance.
Therefore P(polygraph says you’re telling the truth, given that you’re telling the truth) = P(+|T) = 0.61; and
P(polygraph says you’re telling the truth, given that you’re lying) = P(+|N) = 1-P(+|T) = 0.39.
Now using Bayes’ Theorem, we can calculate Chantelois’ chance of evading the lie detector test.
P(Chantelois is not telling the truth, given that the polygraph says she is)
= P(+|N) x P(N) / P(+)
= [ P(+|N) x P(N) ] / [ P(+|T)xP(T) + P(+|N)xP(N) ]
= [ 0.39 x 0.47 ] / [ 0.61 x 0.53 + 0.39 x 0.47 ]
= 0.362 (i.e. 36.2%)
Too high to put any kind of faith in the results of the test.
The calculations above were all done with tongue planted firmly in cheek and are not to be taken seriously. Whether it’s Rann or Chantelois really telling the truth I don’t know or care. What is important is that Bayes’ Theorem shows us that, even with accurate tests, there is a good chance of a misclassification. A single test is usually not enough.