I can't believe a specious argument about conditional probabilities went unnoticed by both the authors and the beta readers. On pp 66-67, Brown and Ladyman write:
"The core of the argument is this: a miracle is a violation of a law of nature. A law of nature has been established on the basis of compelling evidence. It is probable that the violation did not occur and we are obliged to look critically at the evidence for the violation. In an Introduction to Hume’s Enquiry Concerning Human Understanding, Millican gives a striking illustration to make Hume’s case. He asks us to imagine he is worried about a very rare disease that affects only one person in a million, and that there is a test for the disease that produces positive results with 99.9-percent accuracy. That is, out of every 1000 people who test positive for the disease, 999 actually have it and one does not. He then imagines taking the test and getting a positive result. Ought you to conclude from this evidence that he probably has the disease? No, notwithstanding the positive result it is still very much more likely that he is the one in 1000 whose test result is wrong, than the one in a million who actually has the disease." [Emphasis mine.]
Now, anyone with a basic understanding of probability theory should see through this fallacious reasoning. Brown and Ladyman are comparing apples with oranges. Someone with a positive result but with no disease is only one in a thousand. Okay, but one in a thousand what? One in a thousand people with positive feedbacks. In the general populace, are they one in a thousand? Not at all. In fact, they are strictly rarer than people with disease. Here's a mathematical demonstration.
According to Brown and Ladyman,
P(Disease) = 0.000001
P(No Disease) = 0.999999
P(Disease | Test +) = 0.999
P(No Disease | Test +) = 0.001
Using Baye's Theorem,
P(Test +) P(Disease | Test +) = P(Disease) P(Test + | Disease)
0.999 P(Test +) = 0.000001 P(Test + | Disease)
Likewise, P(Test +) P(No Disease | Test +) = P(No Disease) P(Test + | No Disease)
0.001 P(Test +) = 0.999999 P(Test + | No Deases)
Eliminating P(Test +),
P(Test + | No Deases)/ P(Test + | Disease) = 1/111111 < 10^(-6)
P(Test + | No Disease) < P(Test + | Disease) × 10^(-6) < 10^(-6)
This shows that if you don't have disease, there is at most only one in a million chance to get a positive result.
Moreover, P(Test + and No Disease) = P(No Disease) × P(Test + | No Disease) < 10^(-6) × 0.999999 < P(Disease)
This justifies the claim that people with a positive result but with no disease are strictly rarer than people with the rare disease. It is more likely that you have disease if you test positive in this scenario.
Now, Brown and Ladyman cites Millican. Does Millican commit the same fallacy or is he being misrepresented here? It turns out that Millican's argument is being misrepresented by Brown and Ladyman. Millican's argument in his own words is this:
"Suppose, for example, that I am worried about a genetic disease that afflicts one in a million people, and take a test for it which has a 99.9% chance of giving the ‘correct’ result (i.e. if I have the disease, it is 99.9% likely to come out positive, and if I don’t, it is only 0.1% likely to come out positive). Most people would naturally take a positive result as showing that I very probably have the disease. However the one in a million ‘background probability’ outweighs the one in a thousand chance of the test’s getting it wrong, leaving an overall probability that I have the disease, based on this evidence, of only 1 in 1,002. Thus a false test is far more likely than the disease itself."
Mathematically, according to Millican,
P(Disease) = 0.000001
P(No Disease) = 0.999999
P(Test + | Disease) = 0.999
P(Test + | No Disease) = 0.001
Notice that this is not equivalent to saying "Out of every 1000 people who test positive for the disease, 999 actually have it and one does not." Millican says, "If I have no disease, it is 0.1% likely to come out positive" while Brown and Ladyman say, "If I come out positive, it is 0.1% likely that I have no disease." What we see is basically a conflation of conditional probabilities. Brown and Ladyman conflates P(Test + | Disease) with P(Disease | Test +); P(Test + | No Disease) with P(No Disease | Test +).
Otherwise, I would have given this book 3 stars but for this conflation–most likely an oversight–I take out a star.