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Say in a certain population one person is infected with a contagious virus. To prevent the virus from spreading the government tests the entire population whether one individual has this disease or not. The test is 99.9% accurate, but not 100%. Is a certain person, whom is tested positive, infected with the virus? One might say yes, but the test could be wrong.

In fact let's think about this problem logically. When the entire population consists of 10,000 people, chances are that 10 people will get a positive result and in 9 out of those cases it is a false positive. In all of a sudden the 99.9% certainty dropped to 10%...

I won't get too mathematical here. But when a diagnosis is not a hundred percent certain, what can be said about the person having the disease? Does he have the disease or not? Or both? Perhaps both states or possible at the same time?

Raymond Timmermans
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  • https://en.wikipedia.org/wiki/Bayesian_inference. Regarding disease diagnosis, always get a second opinion. –  Apr 05 '17 at 18:36
  • Video on this from just yesterday: https://youtu.be/R13BD8qKeTg – Dave Apr 06 '17 at 19:01

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The answer must be more boring than you would like it to be. "Observations have been made which are consistent with the person having the disease."

In the normal way of thinking about things, you cannot use probabilities to say he has a 99.9% chance of having the disease, for the fact that he has the disease or does not have it is typically assumed to be a statement that has a single truth value. He has the disease, or he does not, we are merely trying to obtain evidence to defend a hypothesis one way or another. When we say "he has a 99.9% chance of having the disease," that is actually just a lazy way of saying "if we were to observe an arbitrary member of our society as having these symptoms, 99.9% of the time that person would have the disease." We have to make sure we're talking about some hypothetical person that doesn't actually exist, or random variables just plain don't work out.

Philosophically, one would state that whether he has the disease or not is not dependent on whether an outside person has observed it.

The issue you mention about false positives is a real one, though it's far more of a mathematical one than a philosophical one. False positives are a huge issue in detecting unusual events, such as terrorist attacks. If you have a detector operating in an airport which detects terrorists and has a false alarm rate of 0.001%, you will still flag 3740 people, of which most likely none are terrorists.

Using Bayesian inference, if you had an existing hypothesis about whether the person had the disease, you could update that hypothesis using the new evidence in a statistically rigorous manner. However, that also is a mathematical approach rather than a philosophical one.

Cort Ammon
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  • (+1) A quibble on your wording: the "...arbitrary..." sentence says that one person is being observed (once); thus how can it have meaning to say "99% of the time"? – rolando2 Apr 07 '17 at 12:06
  • @rolando2 Good point. Would it work better if I changed it to a sample instead of an individual: "if we were to observe an arbitrary sample of members of our society with that disease..." – Cort Ammon Apr 07 '17 at 15:33
  • Maybe, though to asymptotically approach 99.9% you would need that sample to become extraordinarily large. It's difficult to find just the right wording, I know. – rolando2 Apr 07 '17 at 15:45
  • @rolando2 The wording was easier when this was on Philosophy.SE. Here, you guys are a bit more picky about the wording (for good reason, of course!) – Cort Ammon Apr 07 '17 at 16:00