Prevalence of the epidemic: 1/1000.
There are no symptoms.

If there are no symptoms at all and the prevalence is 1 in 1000, then the chance that he is sick is 1 in 100. There is no reason to enter a risk of 1 in 5. This is not a question of Jewish law but of probability.

1 in 10?

Yes. Typo.

How is the chance that he is sick 1 in 10? After all, he has a positive test with 99% reliability. That is, there is a 99% chance that he is sick. In addition, there is a 0.5% chance that the test gave a false negative, meaning there is another tiny chance of 0.5% divided by 1000 that he is nevertheless sick. All in all, the chance that he is sick is over 99%.

Sorry, what I wrote about the additional percentage is not correct. But still, there is a 99% chance that he is sick, since the chance that the test was correct is 99%.

Because the prevalence of the disease is 1/1000. So when you calculate it using Bayes' theorem, the chance that the person is sick given that he has a positive test comes out to 1/10.

Think about a situation where the prevalence of the epidemic is 1 in a billion—do you still think the person is positive with 99% probability?

I understand. Because of the low prevalence in the population, there is a good chance of a false positive. Thanks.

Doesn't this depend on the meaning of "99% reliability"? Does it mean: (a) given a positive test, the chance that you are sick is 99%, or (b) given that you are sick, the chance that you will get a positive test is 99%?

Here the intention is meaning (b), as stated in the question (and also, given that you are healthy, the chance that you will get a negative test is 99%).
This is information that the company producing the drug can determine and publish, and it depends on the quality of the test itself.
The information of meaning (a) can be inferred if one knows meaning (b) and the prevalence of sick people in the population at the given point in time, and therefore it does not depend only on the quality of the test itself.

Okay, thanks. That really isn't clear from the way it was presented.

This is a very nice question, which really shows how unintuitive Bayes' results are.

ףף, Itai is right in his question, and that is what I meant in my question. It really isn't so simple. In the end, the question is what exactly "99% reliability" means.
At first I thought it meant that if you take 100 people, the result will be 99 correct answers and one incorrect one. But I asked someone who is an expert in this field (a professor of neurobiology who deals with public health matters), and he told me that it means you have a 99% chance that all 100 will be negative (because the disease is 1/1000).
Take, for example, 1000 people—what will the results be? Not 990 correct and 10 incorrect. Rather, with 99% probability there will be one positive and 999 negative.
It is still a bit unclear to me too (because those two possibilities do not contradict each other), and therefore I asked the Rabbi for his opinion on whether he is familiar with the topic.

Itai, that is the meaning of the statement about a test's reliability. There is no way to measure the opposite datum; otherwise our situation would be excellent. It is calculated using Bayes. The matter appears at length in several of my columns here. You can search for 'Sir Roy Meadow' or 'Munchausen syndrome by proxy.' Also in my article in Assia, which also appears here on the site.

Rabbi, what about what I wrote? About the two meanings of test reliability.

See my last response. The meaning is always (b). It is impossible to measure reliability in sense (a). But in principle, if you had data about reliability in sense (a), then you would indeed be right and there would be no need to resort to Bayes. Why do people always talk about false positives and negatives? Because they are dealing with reliability in sense (b).